NBER WORKING PAPER SERIES DIVERSIFICATION AND THE OPTIMAL CONSTRUCTION OF BASIS PORTFOLIOS. Bruce N. Lehmann David M. Modest

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1 NBER WORKING PAPER SERIES DIVERSIFICATION AND THE OPTIMAL CONSTRUCTION OF BASIS PORTFOLIOS Bruce N. Lehmann David M. Modes Working Paper 9461 hp:// NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachuses Avenue Cambridge, MA 0138 January 003 We are graeful o he Columbia Business School for providing us wih he compuaional resources o carry ou his analysis, and o Arhur Warga for helpful commens. The research repored here is par of he NBER's research program in Asse Pricing. Any opinions expressed are hose of he auhors and no hose of he Naional Bureau of Economic Research or Morgan Sanley. The views expressed herein are hose of he auhors and no necessarily hose of he Naional Bureau of Economic Research. 003 by Bruce N. Lehmann and David M. Modes. All righs reserved. Shor secions of ex no o exceed wo paragraphs, may be quoed wihou explici permission provided ha full credi including noice, is given o he source.

2 Diversificaion and he Opimal Consrucion of Basis Porfolios Bruce N. Lehmann and David M. Modes NBER Working Paper No January 003 JEL No. G1 ABSTRACT Nonrivial diversificaion possibiliies arise when a facor model describes securiy reurns. In his paper, we provide a comprehensive examinaion of he meris of various sraegies for consrucing basis porfolios ha are, in principle, highly correlaed wih he common facors underlying securiy reurns. Three main conclusions emerge from our sudy. Firs, increasing he number of securiies included in he analysis dramaically improves basis porfolio performance. Our resuls indicae ha facor models involving 750 securiies provide markedly superior performance o hose involving 30 or 50 securiies. Second, comparaively efficien esimaion procedures such as maximum likelihood and resriced maximum likelihood facor analysis (which imposes he APT mean resricion) significanly ouperform he less efficien insrumenal variables and principal componens procedures ha have been proposed in he lieraure. Third, a varian of he usual Fama- MacBeh porfolio formaion procedure, which we call he minimum idiosyncraic risk porfolio formaion procedure, ouperformed he Fama-MacBeh procedure and proved equal o or beer han more expensive quadraic programming procedures. Bruce N. Lehmann Universiy of California, San Diego IR/PS 9500 Gilman Drive La Jolla, CA and NBER blehmann@ucsd.edu David M. Modes Morgan Sanley

3 One of he basic enes of invesmen managemen is ha diversificaion pays: risk can be reduced by spreading invesmens across a large number of imperfecly correlaed asses. The meris of diversificaion are paricularly clear in linear facor models for equiy reurns. These models assume reurns are generaed by a small number of common facors plus an addiional random componen ha can be diversified away in large porfolios. Linear facor models form he basis of he arbirage pricing heory (APT) of Ross (1976, 1977). They are also widely used in quaniaive invesmen managemen for a variey of purposes including risk managemen, alpha deecion, and performance evaluaion. One agnosic and prevalen approach o implemening facor models is o rea he underlying common facors as unobserved random variables, and o infer hem from he covariance srucure of reurns. In his approach, he facor model parameers of individual securiies are esimaed and hen used o form basis or reference porfolios o mimic he common facors. Cross-secional regression based on facor sensiiviy and idiosyncraic risk esimaes remains he mos widely used porfolio formaion procedure. In heory, hese basis porfolio reurns will be boh highly correlaed wih he common facors and relaively free of unsysemaic risk. In pracice, here is no guaranee ha a basis porfolio formaion procedure based on a finie sample will mimic he facors sufficienly well. Performance differences migh be expeced across porfolio consrucion and facor model esimaion procedures and migh well vary wih he number of securiies included and/or he number of facors being considered. I is clearly imporan o know which mehods do a good job of mimicking he common facors and which do no. Which sraegy is bes? This is an empirical quesion ha has no been previously addressed. In his paper, we remedy his omission by providing a comprehensive examinaion of 1

4 differen basis porfolio formaion sraegies. In paricular, we provide a deailed analysis of he performance of varians of all of he porfolio formaion procedures and esimaion mehods ha have been proposed in he lieraure, as well as an examinaion of he performance gains in crosssecions of up o 750 securiies. The paper is organized as follows. The nex secion briefly reviews mulifacor models of asse reurns and heir implicaions for he behavior of diversified porfolios. The hird secion delineaes he differen porfolio formaion mehods considered while he fourh describes he esimaion mehods used o generae he inpus o hese procedures. The fifh secion analyzes he problem of basis porfolio performance evaluaion. The penulimae secion empirically compares and conrass differen esimaion mehods, differen porfolio formaion procedures, and differen numbers of securiies. The final secion provides some concluding remarks. 1. Diversificaion in Mulifacor Models of Asse Reurns A reurn generaing process wihin which i is easy o characerize diversifiable risk is given by he following linear facor srucure: R% = E+ B % δ + % ε ; E[ % δ ] = E[ % εδ% ] = 0; E[ % δ % δ ] = I ; E [% ε % ε ] =Ω (1) K where R % and E are N-vecors of reurns and expeced reurns, respecively, B is an NxK marix of facor loadings,δ % is a K-vecor of mean zero common facors normalized o have a KxK ideniy covariance marix I K, andε% is an N-vecor of mean zero disurbances wih posiive definie covariance marix Ω. The assumpion ha Ω is posiive definie implies ha no asse conains only facor risk. No subsanive assumpions are made in (1) excep for he exisence of he relevan firs and second momens. 1 In paricular, no assumpion has ye been made regarding he poenial o diversify away idiosyncraic risk.

5 Addiional srucure can be placed on (1) by assuming ha he common facors are he dominan source of covariaion among reurns or, equivalenly, ha he residuals represen diversifiable risk. The residuals will be diversifiable if a weak law of large numbers applies o hem. Chamberlain and Rohschild (1983) provide a paricularly convenien sufficien condiion for such a weak law of large numbers o hold: ha he larges eigenvalue of Ω remains finie as N. Chamberlain and Rohschild (1983) ermed (1) coupled wih his consrain on Ω an approximae facor srucure. Wha diversificaion opporuniies arise in a world wih an approximae facor srucure? An arbirary porfolio wih weighs w p has firs and second momens of reurns given by: R% = w R% = w E+ w B% δ + w % ε ; Var[ R% ] = b b + σ p p p p p p p p = E + b % δ + % ε ; σ = w Ωw p p p % ε p p p % ε p () where E p is he porfolio s expeced reurn, b p is a Kx1 vecor of he porfolio s facor loadings, and ε% p is he porfolio s residual reurn a ime. Porfolio p is a well-diversified porfolio if is weighs are of he order 1/N and he sum of squared weighs converges o zero (i.e., w p w p 0). The idiosyncraic risk of a well-diversified porfolio vanishes in he limi when reurns have an approximae facor srucure because σ = w Ωw w w ξ ( ) 0 max Ω where ξ max (Ω) is he ε p p p p p larges eigenvalue of Ω. A well-diversified porfolio wih no exposure o facor risk (i.e., b p = 0) generally exiss unless a vecor of ones approximaely lies in he column span of B. Such a facor-neural porfolio canno be formed from a finie asse menu if he facors can be normalized so ha Bι=ι, where ι is a suiably conformable vecor of ones. A convenien sufficien condiion for he 1 A his sage, hese momens can be eiher condiional or uncondiional. 3

6 exisence of such a porfolio in he infinie dimensional case is ha he covariance marix of he N N j= 1 j j rows of B is nonsingular in he limi. 3 1 Tha is: Var[ B] = ( β β)( β β) is posiive 1 definie where β = N β j and β j is he j h row of B. Noe ha he reurn of such a porfolio (if N j= 1 i exiss) mus converge o he riskfree rae as N or else here is an arbirage opporuniy. The possibiliy of creaing a riskless porfolio from a se of risky asses is one of he main differences beween mean-variance analysis in finie and infinie-dimensional asse markes. The main resricion on he limiing behavior of well-diversified porfolios involves heir expeced reurns. Since he reurns of any K+1 well-diversified porfolios are perfecly correlaed in he limi under hese assumpions, heir limiing expeced reurns mus be linearly dependen as well. Accordingly, here exiss a K-vecor λ and a consan λ 0 such ha E p λ 0 + b p λ for all well-diversified porfolios p. Noe ha his mean resricion is no he APT, a heory ha convers his exac expression for well-diversified porfolio expeced reurns ino an approximae expression for individual asse expeced reurns. Hence, his analysis is free of he conroversy surrounding he APT.. Basis Porfolio Formaion A saisician seeking o esimaeδ % in (), given a priori knowledge of R%, B, E, and Ω, would naurally use generalized leas squares. This is he minimum variance linear unbiased esimaor: In wha follows, all limis will be aken as N, wih N lef implici o avoid noaional cluer. 3 Se Bι = ι + η and consider a uni cos porfolio w p. The sum of is beas is given by wb p ι b pι = w pι+ w pη. Hence, his sum will generically be one if p is well diversified and η can be eliminaed via diversificaion. A column vecor of ones will approximaely lie in he span of B if, for example, one of he common facors is unanicipaed inflaion and all bu a finie number of securiies reurns are equally affeced by his facor. 4

7 ˆ ˆ B B B R% E B B B % Var % B B (3) GLS GLS 1 1 δ = ( Ω ) Ω [ ] = δ + ( Ω ) Ω ε; [ δ δ] = ( Ω ) where B Ω B B Ω % is he error in racking δ. The esimaor ˆ ( ) ε δ GLS is consisen if he smalles eigenvalue of B B grows wihou bound as N, he usual OLS consisency condiion. The rae a which ˆ δ GLS converges hinges on wo facors: (a) he srucure of Ω and (b) he dispersion across he rows of B. Ceeris paribus, convergence is faser when he securiies used in esimaion have smaller idiosyncraic variances, lower correlaions among heir idiosyncraic risks, and greaer differences among heir risk exposures. The las facor is well undersood in a regression seing: precise esimaes of he covariance beween he dependen and independen variables are hard o obain in small samples if he independen variables vary lile across observaions. Similarly, large cross-secions are needed o obain precise facor esimaes if securiies ypically possess similar risk exposures. For example, suppose wo of he facors are unexpeced changes in expeced inflaion and unanicipaed inflaion bu only a small fracion of he asses respond differenly o hem (i.e., inflaion has a neural impac on mos securiy reurns). In his case, accurae esimaion of boh common facors would require a much larger cross-secion han would ypically be needed o merely eliminae idiosyncraic risk. In pracice, he choice among basis porfolio consrucion mehods is furher complicaed by he need o esimae B and Ω. The esimaion of Ω requires he placemen of consrains on is funcional form. Popular choices are he saisical facor analysis model, in which Ω is diagonal, and he principal componens model, in which Ω is a diagonal marix wih equal variances. Measuremen error in he esimaes of B and Ω can impair he small sample properies of basis porfolio esimaors and, as a consequence, large cross-secions may be needed o produce reliable basis porfolio esimaes. 5

8 In wha follows, we consider four procedures for consrucing basis porfolios. Two of he mehods involve biased and unbiased versions of he generalized leas squares esimaor discussed above. Unbiasedness is no a paricular virue, paricularly in he presence of measuremen error and in he absence of a naural scale for he laen facors. Hence, i may be desirable o seek a biased esimaor wih poenially lower variance. The oher wo mehods use mahemaical programming procedures ha consrain he reference porfolios o be welldiversified, which migh miigae some of he harmful effecs of boh sampling error in and he imposiion of consrains on Ω. Before considering alernaive porfolio formaion procedures, i is useful o ranslae his saisical formulaion ino he language of opimal porfolio formaion afer he fashion of Lizenberger and Ramaswamy (1979) and Rosenberg and Marahe (1979). The generalized leas squares esimaor provides wha we usually refer o as Fama-MacBeh porfolios, afer rescaling he porfolios o cos a dollar. 4 In paricular, we choose he N porfolio weighs w j o mimic he j h facor so ha hey: min w Dw subjec o w b = 1; w b = 0 j k (4) wj j j j j j k where b k is he k h column of he sample facor loading marix B and D is he diagonal marix consising of he sample idiosyncraic variances. 5 The soluion is he unbiased minimum idiosyncraic risk porfolio ha mimics he j h unobservable common facor. We rescale he weighs w j so ha he porfolio coss a dollar (i.e., w ι = 1) o mainain comparabiliy wih oher basis porfolio formaion procedures. j 4 Of course, Fama and MacBeh (1973) used he ordinary leas squares esimaor. Our usage is, however, common. 6

9 An alernaive mehod ha produces wha we erm minimum idiosyncraic risk porfolios involves choosing porfolio weighs w j so ha: min w Dw subjec o w ι = 1; w b = 0 j k (5) wj j j j j k In principle, his procedure should produce minimum idiosyncraic risk porfolios whose flucuaions are proporional o he common facor. In conras o he unbiased OLS esimaor, he proporionaliy facor need no equal one. 6 This is easily seen by comparing (4) and (5). I is easy o disinguish hese minimum idiosyncraic risk porfolios from he more familiar Fama-MacBeh porfolios in he one facor case. Assume for simpliciy ha he idiosyncraic variances are idenical (i.e., D = σ ε I N ). In his case, he Fama-MacBeh porfolio solves he programming problem: min w w subjec o w b = 1 (6) w wih soluion w= ( bb ) 1 bprior o rescaling so ha he porfolio weighs sum o one where b is he vecor of sample facor loadings. Similarly, he minimum idiosyncraic risk porfolio saisfies: min w w subjec o w ι = 1 (7) w which is solved by he equally weighed porfolio w = 1 N ι. Thus Fama-MacBeh porfolio weighs are proporional o he individual securiy sample facor loadings (i.e., he beas) and, hus, ake advanage of he differing informaion conen of individual securiies regarding he flucuaions in he common facor. Minimum idiosyncraic risk porfolios, however, are merely well- 5 Noe ha we are now ignoring off-diagonal elemens of Ω such as indusry effecs. As a consequence, our procedures acually are beer characerized as weighed leas squares or diagonal generalized leas squares. 6 Le B = (b 1 b b K ) and suppose we are ineresed in mimicking he j h facor. The minimum idiosyncraic risk esimaor is D -1 B*[B* D -1 B*] -1 where B* = (b 1 b ι b K ) and ι is a vecor of ones in he j h column. 7

10 diversified and do no exploi such informaion. Noe ha he facor loading of he minimum 1 idiosyncraic porfolio is he average bea of he securiies (i.e., b b ι ) while ha of he Fama- MacBeh porfolio is uniy prior o rescaling. The diversificaion properies of Fama-MacBeh porfolios depend on he normalizaion of he common facors. If he facors are normalized so ha facor loadings are ypically close o one, 7 boh procedures yield porfolio weighs of order 1/N. However, Fama-MacBeh porfolios have very large weighs in finie cross-secions when average facor loading esimaes are on he order of o as is ypical in daily daa under he normalizaion Var( δ ) = I. This is = N % K no a problem when he facor model parameers are measured wihou error bu i is a poenially serious source of difficuly when large facor loadings can reflec measuremen error as well as responsiveness o common facors. The comparaive meris of minimum idiosyncraic risk and Fama-MacBeh porfolios in he presence of measuremen error can be invesigaed more fully by again considering he one facor model: R% E % % E % % I (8) = + βδ + ε; [ εε ] = σ ε N; σ β > 0 where β βι is he N x 1 vecor of he (no idenical) populaion facor loadings and he idiosyncraic disurbances are assumed, for simpliciy, o have zero means, common (known) variances σ ε, and o be disribued independenly of one anoher. Suppose ha we measure he facor loadings β wih error: b= β + υ; E[ υ] = 0; E [ υυ ] =Συ (9) 7 We are no aware of any sudy in which he facors are normalized o ensure ha he ypical loading is uniy. The following ransformaion yields ypical loadings of one. Transform B so 8

11 where υ is an N vecor of he deviaions of he sample facor loadings from heir populaion counerpars wih υ assumed o be independen ofδ % and % ε. 8 We also normalize he facor loading esimaes such ha b b = b ι. This normalizaion implies ha 0<b <1 o ensure ha he sample variance of he b i s is posiive. The Fama-MacBeh procedure yields a basis porfolio wih a sample loading of uniy and wih weighs ha sum o one. Finally, we assume ha he 1 cross-secion is large enough o ensure ha b β = N N β j 1 j [i.e., 1 N υ 0 j 1 j ] o simplify = N = he arihmeic. In his seing i is easy o characerize he behavior of he Fama-MacBeh porfolio and he relaions beween is reurns and he common facor. The porfolio reurns are: R% = bb b E+ % + % = bb E+ % + % + E+ % + % FM 1 1 ( ) [ βδ ε] ( ) [ β ββδ β ε υ υβδ υε] 1 [ β E+ β βδ% + β % ε + υ + υ βδ% + υ % E ε ] N β (10) where he approximaionb β implies ha Nβ bb. The variance of Fama-MacBeh porfolio reurns as well as heir squared correlaion wih he common facor are (approximaely) given by: 1 Var R% E E r N β FM [ ] [( ββ + βσ ) ( )] υβ σδ + ββσε + Σ υ + σε Συ [ % FM Corr R, % δ ] ββ σ ( ββ β β) σ ββσ σ ( ) δ + Σ υ δ + ε + EΣ υe+ εr Συ (11) ha B D -1 B is diagonal and denoe he j h diagonal elemen as γ j. Then B*=BA where A is a diagonal marix wih ζ j /γ j along he diagonal where ζ=b D -1 ι has a ypical loading of one. 8 We assume ha our esimaes are unbiased for simpliciy. The assumpion ha υ is independen of % δ andε% is less innocuous since we ypically esimae facor loadings and form basis porfolios on he same sample. Accouning for such problems would complicae he analysis considerably wihou alering he basic poin. For example, if b were esimaed from an ordinary leas squares δσε δσε regression, hen cov[ υ, % ε ] = T = would ypically be quie small in moderaely large samples. i δ T [ σ ] 1 δ + δ % = 9

12 whereσ is he variance of he common facor. δ I is easier o evaluae he corresponding quaniies for he minimum idiosyncraic risk porfolio because i ignores he informaion in he beas. Hence, is reurns and heir corresponding momens are: % MIRP 1 [ ] ; [ ] σ = ι + βδ% + ε = + βδ% + ε % MIRP R % E E Var R β σδ + N N βσ [ % MIRP Corr R, % δ] βσ / N δ δ + σε ε (1) Wha are we o make of his edious arihmeic? The laen naure of he facors makes heir scale arbirary, making high correlaion he appropriae objecive as opposed o low racking error. When β is measured wihou error [i.e., υ = 0], he Fama-MacBeh porfolio reurn is more highly correlaed wih he common facor han ha of he minimum idiosyncraic risk porfolio since: ( σ + β ) σ Corr[ R%, ] [ %, ] (13) FM % ββσ δ β δ β σδ MIRP δ = % Corr R δ ββσ δ + σε ( σβ + β ) σδ + σε / N β σδ + σε / N N j= where 1 σ ( ) 0 β = N β β > 1 j is he sample cross-secional variance of he populaion beas. The same canno be said when here is sampling error in he beas: he squared correlaion of minimum idiosyncraic risk porfolio reurns wih he common facor is unaffeced by measuremen error in b while ha of he Fama-MacBeh porfolio falls in he presence of sampling error. For example, consider he special case in which b is esimaed by ordinary leas squares regression of % R E on % δ, in which case he error covariance marix is (approximaely): σ Συ Tσ ε δ I N (14) 10

13 where T is he sample size and he approximaion arises because we replaced he sample mean and variance ofδ % wih heir populaion values for ease of exposiion. The squared correlaion of he Fama-MacBeh porfolio wih he common facor is hen: Corr[ R% FM, % δ ] ββ σ ββ σ β βσ ββσ σ δ δ + Σ υ δ + ε + E Σ υe + εr Σ υ ( ) 1 1 ; σ = ( ) σ ε T + 1 ( σ σ ) E + E + ε N T( σβ + β ) σδ T( σβ + β ) σδ N E Ej E N j= 1 (15) which will be smaller han ha of he minimum idiosyncraic risk porfolio when: T T E (16) ( σ β + β ) σ δ < ( + 1) β ( σ β + β ) σ δ + β ( σe + + σε ) which can easily obain when he sampling variaion in he beas is small or, equivalenly (given he normalizaion 1 = N bb band he approximaion b β ), when β is close o one. Noe ha he variances of sample beas compued wih respec o he usual marke proxies are ofen quie small. While boh of hese procedures produce well-diversified basis porfolios in he limi, hey may no do so in finie asse menus. This possibiliy led Chen (1983) o employ (proprieary) mahemaical programming mehods o produce well-diversified basis porfolios wih a finie number of securiies. Such porfolios migh suffer only marginally from errors in esimaing he facor loadings and idiosyncraic variances due o he diversificaion consrain. Accordingly, our oher wo porfolio formaion procedures involve quadraic programming subjec o fixed upper and lower bounds as in: min w Dw subjec o w ι = 1; w b = 0 j k; l w u (17) wj j j j j k j j j where l j and u j are he fixed lower and upper lower bounds. We examined wo choices for hese bounds. Following Chen (1983), we produced porfolios wih non-negaive weighs (i.e., l j = 0) 11

14 ha could ake on maximum values of one o wo per cen. We also examined porfolios ha were merely consrained o be well-diversified wih weighs aking on maximum and minimum values of plus and minus one or wo per cen. We experimened wih consrains as large as five per cen in absolue value and he resuls did no differ maerially from he ones presened below and, hence, are no repored. Finally, we formed sample facor neural porfolios for each porfolio formaion mehod. These porfolios are used o consruc he excess reurn basis porfolios analyzed below. The differences among excess reurn porfolios are also bes illusraed in he one facor case wih idenical and uncorrelaed idiosyncraic risks. In his case, he required orhogonal porfolio solves he programming problem: 1 σβ + β β min w R w 1; 0 f R subjec o w f R ι = w f R β = w f R = ι β w f R f N σβ σβ (18) The corresponding excess reurn porfolio weighs are obained by subracing hese weighs from w yields: = 1 N ι, he minimum idiosyncraic risk porfolio in his case as noed in (7) above. This w 1 1 σβ + β β β w = ι ι β ( β ιβ ) = N N σβ σβ Nσβ MIRP R f (19) The Fama-MacBeh excess reurn porfolio solves he programming problem: ex 1 min ex ex ex ι 0; ex w = β = 1 = ( β ιβ ) ex FM wfm subjec o wfm wfm wfm wfm Nσ β (0) These manipulaions yield one nonrivial insigh he minimum idiosyncraic risk procedure produces weighs for excess reurn porfolios ha are proporional o he Fama- MacBeh excess reurn porfolios. No surprisingly, he facor of proporionaliy is he average facor loading β. Once again, when he average facor loading is much less han one, he Fama- 1

15 MacBeh procedure will produce very large posiive and negaive porfolio weighs. In conras, he minimum idiosyncraic procedure yields a well-diversified excess reurn porfolio. A similar resul arises in he muliple facor case. The choice of basis porfolio formaion procedure should be a second order consideraion since well-diversified porfolios possess no idiosyncraic risk in he limi and so any K imperfecly correlaed well-diversified porfolios should suffice. In his secion, we have argued ha he consrucion of reliable basis porfolios in pracice requires a large enough asse menu so as o virually eliminae idiosyncraic risk and an asse menu wih sufficienly differen facor sensiiviies so as o allow saisical discriminaion among he facors. Furher, we have shown ha asympoically efficien basis porfolio formaion procedures ha make use of sample differences in facor sensiiviies may have inferior small sample properies o hose mehods ha simply emphasize diversificaion and orhogonaliy. 3. Esimaion Mehods The choice among esimaion mehods involves differen radeoffs han he choice among basis porfolio formaion procedures. The radeoff here is beween saisically efficien bu compuaionally cosly mehods such as facor analysis, and less efficien bu less cosly mehods such as insrumenal variables or principal componens. The quesion is wheher comparaively inefficien mehods provide performance comparable o ha of compuaionally burdensome efficien esimaion mehods. When he facors are no observed, esimaes of B and Ω may be obained from he covariance srucure of reurns, Σ = BB + Ω. When reurns are normally and independenly disribued ha is, whenδ % andε% are joinly normal and serially independen he higher momens of reurns conain no informaion abou hese parameers. In hese circumsances, he 13

16 log likelihood funcion is given by: L NT T T 1 ( Σ S) = lnπ ln Σ ( R % R) Σ ( R % R) = 1 NT T T = π Σ r S Σ 1 ln ln ( ) 1 (1) 1 T since he sample covariance marix S = ( % )( % T R R R R) follows a Wishar disribuion. =1 Unforunaely, he esimaion of B and Ω when securiy reurns possess an approximae facor srucure requires he imposiion of furher consrains on Ω. The wo main alernaives are facor analysis and principal componens: Ω is a diagonal marix D in he former while Ω= σ ε N I in he laer. This observaion highlighs he inuiive disincion beween facor analysis and principal componens facor analysis implicily involves weighed leas squares esimaes of he facors and facor loadings (where he weighs are he esimaed idiosyncraic variances) while principal componens provides he corresponding ordinary leas squares esimaes. Hence, facor analysis should end o perform beer han principal componens when here is nonrivial cross-secional variaion in idiosyncraic variances. We used wo relaively inexpensive echniques for esimaing he saisical facor analysis model. The firs is a modified version of he EM algorihm as applied o facor analysis in Lehmann and Modes (1988), which used an ieraive mulivariae regression procedure. The firs wo seps of he EM algorihm in his paper are unchanged bu he hird sep is new: 1. Given B and D, compue he minimum variance biased esimaor of he facors via ˆ BGLS δ = ( I + B Ω B) B Ω [ R % 9 - E]; 9 ˆ δ BGLS has lower mean squared error han (3) since i uses he informaion in he normalizaion Var( % δ ) = I. K 14

17 . Esimae B and D via mulivariae regression of R % on ˆBGLS δ and 3. Compue he MLE of B given D from sep via B = D ½ Γ(D,S*)Λ -½ (D,S*) where Λ(D,S*) is a diagonal marix comprised of he K larges eigenvalues of S* = D -½ SD -½ and Γ(D,S*) are he associaed eigenvecors. Reurn o sep 1 if he algorihm has no converged. This revised EM algorihm usually converges much more rapidly han he convenional EM algorihm. 10 We refer o his below as unresriced maximum likelihood. The second echnique, which we refer o as resriced maximum likelihood, makes use of he informaion on B implici in he vecor of sample mean securiy reurns when he APT is rue. In his case, expeced securiy reurns are linear combinaions of he produc of heir facor loadings and a se of facor risk premiums. In order o exploi his informaion, we also performed maximum likelihood facor analysis subjec o his consrain by maximizing: NT T T 1 T 1 L ( B, D, λ0, λ S) = lnπ ln Σ r( SΣ ) ( R ιλ0 Bλ) Σ ( R ιλ0 Bλ) () where λ is he vecor of facor risk premiums and λ 0 = 0 if a column of ones lies in he column span of B. We used he unresriced esimaes of B and D from maximum likelihood facor analysis as saring values and hen obained hese resriced maximum likelihood esimaes using a varian of he EM algorihm. Principal componens has been advocaed by Chamberlain and Rohschild (1983) and 10 We found i necessary o creae his modificaion because he convenional EM algorihm spen a grea deal of ime slighly refining he facor loading esimaes in he neighborhood of he maximum of he likelihood funcion. Jones (001) uses a very similar ieraive principal componens procedure ha differs in how he D esimae is updaed. He shows ha he resuling facor esimaes are consisen as N when B is ime invarian if he facors are serially correlaed and he residuals are heeroskedasic and provides a mehod for dealing wih missinga-random daa along he lines of Connor, Korajczyk, and Uhlaner (00). See Lehmann (199) for relaed resuls. 15

18 Connor and Korajczyk (1986,1988) as an inexpensive alernaive o maximum likelihood facor analysis. This mehod requires he exracion of he K larges eigenvalues of S and heir associaed eigenvecors or, equivalenly, he K larges singular values of he marix of reurns and heir corresponding singular vecors. We exraced all of he eigenvalues and eigenvecors of S and scaled each eigenvecor by he square roo of he corresponding eigenvalue o se % K Var( δ ) = I. We hen esimaed D by subsiuing he ransformed eigenvecors associaed wih he K larges eigenvalues for B ino he model Σ = BB + D. 11 Finally, insrumenal variables is anoher inexpensive alernaive o maximum likelihood facor analysis, a varian of which was employed in Chen (1983). The basic idea is quie simple: subsiue consisen esimaes of he facors for he facors hemselves in (1) and esimae he facor loadings and idiosyncraic variances by ordinary leas squares regression. Following Madansky (1964) and Hagglund (198), we normalize he facors so ha I K is he facor loading marix of he firs K securiies, making are he corresponding facors correlaed, and decompose he vecor of reurns ino hree componens: r% = R% E = % δ + % ε r% = R% E = γ % δ + % ε r% = R% E =Γ % δ + % ε (3) where r%, r%, and r% are he demeaned reurns on he firs K, he K+1 s, and he las N K 1 asses, 1 3 respecively, and γ and Γ3are he corresponding facor loading vecor and facor loading marix of he K+1 s and he las N K 1 asses, respecively. Consider he regression of r% on r % 1 : 11 Connor, Korajczyk, and Uhlaner (00) avoid he comparaively expensive eigenvalue/eigenvecor decomposiion by using a wo-pass cross-secional regression mehod ha is a close cousin of he EM algorihm. They also consider mehods for dealing wih daa ha is missing-a-random. 16

19 r% = γ r% + u% = γ r % + % ε γ % ε (4) Clearly applicaion of ordinary leas squares will lead o biased and inconsisen esimaes of γ since r% 1 is correlaed wih is own idiosyncraic disurbance erm. If insead we firs % regress r% 1 on r 3 : r% =Π r% + u % (5) and hen replace r% 1 in equaion (4) wih he fied values from his regression, ordinary leas squares can hen be used o esimae γ consisenly. This esimae is consisen because he fied values from (5) are purged of ε% 1 and he idiosyncraic disurbancesε% 3 are uncorrelaed or sufficienly weakly correlaed wihε% by assumpion. Repeaed applicaion of his procedure afer swapping r % wih each elemen of r % 3 yields consisen esimaes of he corresponding rows of Γ 3. Finally, soluion of he marix equaions: Γ ( S ΓΦΓ D) Γ= 0 Diag( S ΓΦΓ D) = 0 (6) yields esimaes of he facor covariance marix Φ and he idiosyncraic variances D. The Φ esimae can be used o ransform Γ ino B by implicily rescaling he facors o be uncorrelaed wih uni variances. 4. Basis Porfolio Comparison I would be a simple maer o deermine which combinaion of basis porfolio formaion procedure and esimaion mehod works bes if he facors were observed. Of course, we would no need o consruc basis porfolios if we observed he common facors. One heurisic approach o comparing basis porfolios is o examine he behavior of heir weighs and he sample means and variances of heir reurns in order o check wheher he resuls appear o be reasonable. For 17

20 example, basis porfolios ough o be well-diversified in order o diminish idiosyncraic risk. The quadraic programming porfolios are well-diversified by consrucion bu he same need no be rue of Fama-MacBeh or minimum idiosyncraic risk porfolios in finie samples. Similarly, he sample momens of basis porfolio reurns can reveal peculiariies in heir performance. For example, we have examined basis porfolios wih mean reurns and sandard deviaions as high as 10 percen and 450 percen per monh, respecively. Good basis porfolios probably do no exhibi such behavior. Unforunaely, searching for reasonable reference porfolios in his fashion is no likely o eliminae many candidaes. Forunaely, some plausible large sample approximaions faciliae he comparison of well-diversified basis porfolios. The sample mean vecor and covariance marix of R % p are given by: T T T R = R% = ιλ + B λ + ε ; δ = % δ ; λ = ( λ+ δ ); ε = % ε p p 0 p p p T = 1 T = 1 T = 1 T 1 S = ( R% R )( R% R ) = B S B + B S + S B + S p p p p p p δ p p δεp δεp p εp T = 1 T T T S = ( % δ δ )( % δ δ ); S = ( % δ δ )% ε ; S = % ε % ε δ δεp p εp p p T = 1 T = 1 T = 1 (7) where λ 0 is zero if a vecor of ones (approximaely) lies in he column span of B and is he riskless rae oherwise. Sδε p, he marix comprised of he sample covariances beween he acual facor realizaions and he idiosyncraic reurns of he basis porfolios, should be close o is probabiliy limi of zero in large samples (i.e., S p BpSδBp + S ε p ). Similarly, he sample mean vecorε p will be close o zero as well if he basis porfolios are large, well-diversified, and consruced so ha heir weighs are no sysemaically relaed o heε% p realizaions (i.e., Rp ιλ0 + Bpλ). We do no assume ha Sε p is close o zero since ε p will converge o zero faser 18

21 han Sε p for well-diversified porfolios of large numbers of securiies. Now consider he usual χ saisic for esing he hypohesis ha he K mean excess reurns of he basis porfolios over λ 0 are all zero: T( R ιλ ) S ( R ιλ ) Tλ B S B λ ; S = S S B ( S + B S B ) B S p 0 p p 0 p p p p εp εp p δ p εp p p εp Tλ [( B S B ) + S ] λ; B S B = [( B S B ) + S ] p εp p δ p p p p εp p δ (8) The srucure of S p simplifies maers considerably and he χ saisic is (approximaely) bounded by he maximum aainable value of 1 Tλ λ. When B S B is large (i.e., when 1 S δ p ε p p ξ is large), ( 1 ) min BpSε B p p [( BS B) + S] is small and he χ saisic will be large as well p εp p δ Ceeris paribus, reducions in he idiosyncraic variances and increases in he facor loadings of he basis porfolios will increase he percenage of basis porfolio reurn variance explained by % δ and, hus, he χ saisic. Hence, he usual χ saisic for esing he join significance of mean excess basis porfolio reurns can rank differen combinaions of porfolio formaion procedures and facor loading esimaion mehods. For example, in he one facor case discussed earlier, he χ saisics for he Fama- MacBeh and minimum idiosyncraic risk porfolios wih no measuremen error in β are (approximaely): χ χ FM MIRP ββ λ λ T = T ββ σδ + ββσ ε σ β σ δ + N β ( σ β ) β λ T = T σ βσ N λ σ Nβ ε ε δ + σδ + ε β + (9) Tha is, he Fama-MacBeh porfolio performs beer as long as σ > 0. However, he inequaliy β 19

22 can be reversed when here is measuremen error in b since: χ FM T T ββ λ 1 σ ββ σ (1 ) ββσ ( ββ λ σ ) ε δ + + ε + + N ε T Tσ δ λ σ ε β 1 N σδ + (1 ) [1 ( )] λ σ β + β Nβ ( σβ + β ) T Tσδ (30) 1 N This saisic will be smaller han χ MIRP if σ [1 ( β < β + + λ σ )], β + β T Tσ δ which will generically occur for N sufficienly large. Sampling error can indeed cause minimum idiosyncraic risk porfolios o ouperform heir Fama-MacBeh counerpars. This reasoning poins o a problem wih esing he APT afer using he χ saisic o assess basis porfolio performance. Consider he fied mulivariae regression % % p of R ιλ0 on R ιλ0 and a consan: R% ιλ = ˆ α + Bˆ ( R% ιλ ) + ˆ ε (31) 0 p 0 where ˆ α is he vecor of esimaed inerceps, ˆBis he esimaed facor loading marix, and ˆε is he fied residual vecor. The populaion value of α will be zero if he APT is rue and he basis porfolios measure he common facors wih negligible error. The usual χ for esing his hypohesis is: T ˆ α Σ ˆ ˆ α = T[ R ιλ Bˆ( R ιλ )] Σˆ [ R ιλ Bˆ( R ιλ )] 1 1 ε 0 p 0 ε 0 p 0 = T[( R ιλ ) S Σˆ Σˆ Σˆ S ( R ιλ )] ε ε ε 0 = T( R ιλ ) S ( R ιλ ) T( R ιλ ) S ( R ιλ ) p 0 p p 0 (3) where Σ ˆ ε is he sample residual covariance marix of ˆ ε. The las equaliy follows direcly from hree observaions: (1) α =Σ ˆ () -1 ˆ ε S R; ˆ 1 ωp( ω p ωp) ; 1 B= S S and (3) Σ ˆ ε = S Sω p( ω psωp) ωps 0

23 where ω p is he NxK marix of porfolio weighs of he rue basis porfolios. This represenaion makes economic sense: he difference in squared Sharpe raios measures he performance improvemen in mean-variance space arising from he addiion of he individual securiies o he basis porfolios, which will be zero, apar from sampling error, if α = 0. 1 Thus he χ saisic for esing he null hypohesis α = 0 is he difference beween he χ saisics for esing he join significance of wo ses of mean excess reurns, hose of individual securiies and hose of he basis porfolios. Choosing he basis porfolios wih he larges χ saisic minimizes he es saisic for α = 0 and hus reduces is power, alhough he magniude of he bias when he APT is false canno be analyzed wihou furher assumpions. Forunaely, his problem can be miigaed o a considerable exen by using known empirical anomalies such as hose associaed wih marke o book, firm size and dividend yield o increase he power of ess of he APT since basis porfolios end o be well-diversified while he anomalies are no. Tha is, characerisics such as small marke o book or firm size, zero dividend yield, or high dividend yield are clearly no disribued uniformly over securiies. Finally, i is worh emphasizing a major limiaion of he analysis. I is no possible o es analyically wheher he differences in χ saisics are significan a convenional levels because hey are dependen across porfolio formaion and esimaion procedures. 13 Hence, he resuls ha follow should be hough of as suggesive ones o be weighed along wih oher evidence such as he sensiiviy of absolue and relaive muual fund performance measures o alernaive basis porfolio consrucion mehods as documened in Lehmann and Modes (1987). 5. Empirical Resuls Evaluaion of basis porfolio performance requires consideraion of he frequency of 1 This was firs noed in Jobson and Korkie (198). 1

24 observaion, he sample period, he asse menu, and he number of posulaed facors. The main virue of daily daa is he increased precision of second momen esimaes while he biases in mean reurns associaed wih bid-ask spreads and he biases in second momens arising from non-rading and hin rading consiue heir principal defecs. As in mos invesigaions of he APT, we oped for he puaive virues of a large sample and used daily daa o esimae he facor loadings and idiosyncraic variances. We esimaed facor models for four subperiods covered by he CRSP daily reurns file: 1963 hrough 1967, 1968 hrough 197, 1973 hrough 1977, and 1978 hrough 198. In each period, we confined our aenion o coninuously lised firms in order o have he same number of observaions for each securiy and ignored any poenial selecion bias associaed wih his choice. This yielded samples of 1001, 1350, 1350, and 1346 securiies and 159, 134, 163, and 164 daily observaions in hese four subperiods, respecively. To guard agains any biases induced by he naural progression of leers (General Dynamics, General Elecric, ec.), we randomly reordered he securiies in each subperiod. We also made choices as o he number of securiies and he number of facors included in he analysis. In order o sudy he impac of he size of he asse menu on reference porfolio performance, we esimaed facor models for he firs 30, 50, and 750 securiies in our randomly sampled daa files for each period afer seeing lile improvemen in experimens wih larger cross-secions. We chose o remain agnosic abou he rue number of facors and examined models conaining five, en, and fifeen facors, alhough for obvious reasons we did no esimae a fifeen facor model wih only hiry securiies. We also normalized he models so ha he facors were uncorrelaed wih uni variances and ha B'D -1 B was diagonal. This is he sandard 13 The boosrap could be employed here under suiable regulariy condiions.

25 normalizaion in he lieraure, alhough i is worh noing ha Fama-MacBeh porfolios are no invarian wih respec o normalizaion. We firs provide evidence on he diversificaion properies of alernaive combinaions of Fama-MacBeh and minimum idiosyncraic risk porfolios and esimaion mehods, omiing he quadraic programming porfolios since hey are well-diversified by consrucion. There are several ways o quanify he diversificaion properies of reference porfolio weighs. We confine our aenion o one simple summary measure: he sum of squared porfolio weighs, which converges o zero as he number of asses grows wihou bound when a porfolio is welldiversified. The minimum sum ha can be aained wih porfolio weighs ha sum o one is he inverse of he number of securiies: for hiry securiies, for 50 securiies, and for 750 securiies. Table 1 summarizes he diversificaion properies of Fama-MacBeh and minimum idiosyncraic risk porfolios across esimaion mehods, numbers of asses, and facors. For each combinaion, we repor wo numbers: he average sum of squared porfolio weighs across boh facors and sample periods and he sample sandard deviaion of he sum of squared porfolio weighs. These quaniies are given by: ww w w w Varww w w ww T K T K N T K k k = k k = ; ( ik k k ) = ( k k k k ) T = 1 K k= 1 TK = 1 k= 1 i= 1 TK = 1 k= 1 (33) where k indexes facors, refers o ime periods (where T=4), and i indexes firms. Obviously, hese measures are descripive and are no appropriae for inference wihou furher assumpions. The overwhelming message of Table I is ha Fama-MacBeh porfolios formed under he usual facor model normalizaion are exremely poorly diversified. In conras, minimum idiosyncraic risk porfolios proved o be quie well-diversified wih mean sums of squared weighs only en o weny imes he minimum aainable ones. The conras is sriking Fama- 3

26 MacBeh porfolios yielded values of ww k k beween 40 and 50,000 imes hose of minimum idiosyncraic risk porfolios. Examinaion of he resuls for individual basis porfolios (no repored here) reveals ha similar differences occur in he disaggregaed daa as well. In fac, every minimum idiosyncraic risk porfolio was beer diversified han is Fama-MacBeh counerpar. The evidence is less clear on he comparaive performance of differen esimaion mehods. Minimum idiosyncraic risk porfolios formed from principal componens esimaes appear o be slighly beer diversified han hose compued from he oher hree esimaion mehods bu he difference is sligh and is reversed ofen when individual basis porfolios are compared. Wha accouns for he sharp conras in he diversificaion behavior of hese wo porfolio formaion mehods? As noed earlier, he answer is scaling: Fama-MacBeh porfolios are he minimum idiosyncraic risk porfolios wih a loading of one on one facor and loadings of zero on he oher facors prior o rescaling o cos a dollar. Facor loading esimaes as usually normalized are ypically much smaller han one (on he order of o in daily daa) and, hence, some weighs mus be very large and posiive o insure a porfolio loading of one on he facor being mimicked while ohers mus be large and negaive in order o have loadings of zero on he oher facors. This problem does no arise in he CAPM conex since he naural bea on a marke proxy is one. By conras, minimum idiosyncraic risk porfolios need no have any paricular loading on he facor being mimicked and hey need only have small posiive and negaive weighs o insure orhogonaliy o he oher facors. The remainder of his secion is devoed o he evaluaion of alernaive mehods using he χ saisic (8). Tables hrough 4 conain aggregae χ saisics summed over he four subperiods, a valid procedure since sums of independen χ saisics are disribued χ as well 4

27 and he subperiod saisics are independen by assumpion. Few nonrivial differences in performance are obscured by aggregaion and we summarize any relevan ones in he ex. Table provides he fied χ saisics based on daily daa and five, en, and fifeen facor models for various combinaions of porfolio formaion procedures, esimaion mehods, and number of securiies in he asse universe. Table 3 presens he same saisics based on daily reurns from he subsequen five-year period o guard agains any overfiing caused by using he same sample o esimae he facor models and o compue he χ saisics. 14 These saisics are based on a smaller number of securiies since no all securiies were coninuously lised during boh five year periods. Table 4 conains he same saisics compued from weekly reurns in order o miigae any bid-ask spread bias ha migh inflae mean basis porfolio reurns. Each able conains a pleniude of informaion. Panels A, B, and C repor on 5, 10, and 15 facor models, respecively. The four columns lised a he op of each able correspond o he four porfolio formaion mehods under consideraion: minimum idiosyncraic risk, Fama- MacBeh, posiive weigh quadraic programming, and well-diversified quadraic programming. Eigh combinaions of esimaion mehods and numbers of securiies comprise he rows of each able: maximum likelihood and resriced maximum likelihood facor analysis wih 30, 50, and 750 securiies and principal componens and insrumenal variables wih 750 securiies. The χ saisic for he null hypohesis ha he raw and excess mean reurns of he K basis porfolios are all zero along wih he associaed marginal significance level (in parenheses underneah i) is repored for each porfolio formaion mehod, esimaion mehod, and number of securiies. The excess reurns are he difference beween raw reference porfolio reurns and minimum idiosyncraic risk orhogonal porfolio reurns. Noe ha he χ saisics for he excess reurns of 14 Hence, here are no resuls for he firs five year period. 5

28 he minimum idiosyncraic risk and Fama-MacBeh porfolios are idenical as noed earlier. 15 The resuls srongly sugges he imporance of using large cross-secions o consruc basis porfolios. The 750 securiy basis porfolios always ouperformed heir 50 securiy counerpars which, in urn, always dominaed 30 securiy porfolios. The ypical χ saisics for 50 and 750 securiies were roughly wice and hree imes as large, respecively, as hose for 30 securiies. These rankings hold across esimaion and porfolio formaion procedures, ime periods, and observaion inervals for boh raw reurn and excess reurn porfolios (no all of which are repored here). These differences increase wih he number of facors in sample in he daily daa, alhough his finding does no persis in he weekly daa or ou of sample. Noe ha his need no have occurred we have found randomly seleced 50 securiy porfolios wih larger χ saisics han hose of 750 securiies. For example, he χ saisics for he minimum idiosyncraic risk basis porfolio mean excess reurns based on unresriced maximum likelihood facor analysis were 35.0, 81.14, and using 30, 50, and 750 securiies, respecively. The mean reurns of he 30 securiy raw reurn basis porfolios were insignifican a he en per cen level for wo ou of four subperiods in daily daa while he corresponding mean excess reurns were insignifican in hree ou of four periods. In conras, he 50 and 750 securiy raw reurn porfolios were highly significan a convenional levels in all subperiods while he excess reurn porfolios were highly significan in daily daa for all bu he five facor model in he hird period. The four esimaion mehods exhibied similarly sriking conrass. Resriced maximum 15 Recall ha he Fama-MacBeh and minimum idiosyncraic risk excess reurn porfolio weighs are proporional where he facor of proporionaliy is he inverse of he average facor loading in he one facor model. Fama-MacBeh porfolio weighs end o be much larger since he average loading is ypically much smaller han one. However, he proporionaliy facor cancels ou in he consrucion of he χ saisic. 6

29 likelihood facor analysis sysemaically ouperformed is unconsrained counerpar in daily and weekly daa bu his dominance did no persis ou of sample, where hey achieved almos idenical performance. Boh maximum likelihood procedures consisenly ouperformed he less efficien insrumenal variables and principal componens mehods. Consider, for example, en facor minimum idiosyncraic risk excess reurn porfolio resuls in daily daa in sample: he χ saisics based on resriced and unresriced maximum likelihood facor analysis, principal componens, and insrumenal variables were 143.9, , 97.88, and 8.94, respecively. Perusal of he ables indicaes ha he differences in χ saisics across esimaion mehods were ypically much larger for en and fifeen facor models han for five facor models. The superioriy of he relaively efficien maximum likelihood procedures also persised ou of sample and in weekly daa. The four esimaion mehods generaed χ saisics of 53.8, 53.50, and ou of sample and 90.33, 73.83, 53.08, and 6.48 in weekly daa, respecively. An examinaion of he subperiod resuls mosly suppors his conclusion, albei wih some ineresing ineremporal variaion in performance. Principal componens performed well in-sample only in he firs five year period while achieving performance comparable o he 50 securiy basis porfolios in he subsequen hree periods. Insrumenal variables provided more consisen performance han principal componens in-sample bu was consisenly inferior o he more efficien esimaion procedures. The ou-of-sample resuls were broadly consisen wih hese findings wih some idiosyncrasies in he individual subperiods. In five facor models, he efficien mehods proved superior in he second subperiod while all mehods yielded similar performance in he firs and hird subperiods. In en facor models, all mehods provided similar performance in he firs and second subperiods excep for he superior performance of insrumenal variables in he firs and is inferior performance in he second. The efficien 7

30 mehods ouperformed he inefficien mehods by a wide margin in he final ou-of-sample period. Similar inconsisencies emerged in he fifeen facor runs. The mehods provided similar performance in he firs ou-of-sample period excep for he inferior resuls provided by principal componens. In he second subperiod, only insrumenal variables provided inferior resuls o he similar performance of he oher hree mehods. Again, he efficien mehods ouperformed he inefficien mehods in he final subperiod. The final comparisons are among he four porfolio formaion mehods. Minimum idiosyncraic risk porfolios provided almos idenical performance o well-diversified quadraic programming porfolios using 750 securiies and provided consisenly superior performance wih 50 securiies. Oher observaions depend on he esimaion mehod and choice of raw or excess reurns. 16 For he efficien esimaion mehods and raw reurns, minimum idiosyncraic risk porfolios consisenly dominaed Fama-MacBeh porfolios in-sample in boh daily and weekly daa which, in urn, ypically ouperformed he posiive weigh quadraic programming porfolios. For insance, he daily in-sample χ saisics for five facor minimum idiosyncraic risk and Fama-MacBeh raw reurn porfolios based on maximum likelihood facor analysis were and 9.8, respecively. The corresponding numbers for en facors were 01.1 and , respecively, and for fifeen facors were 7.77 and 168.4, respecively. However, Fama-MacBeh porfolios slighly ouperformed minimum idiosyncraic risk porfolios ou-ofsample while boh coninued o dominae he posiive weigh quadraic programming porfolios. The picure regarding he relaive meris of he minimum idiosyncraic risk, Fama-MacBeh, and quadraic programming excess reurn basis porfolios was virually idenical for he efficien 16 As noed above, he minimum idiosyncraic risk and Fama-MacBeh porfolio formaion procedures yield idenical χ saisics for excess reurn basis porfolios. However, he χ saisics 8