Investing on the CAPM Pricing Error

Transcription

1 Tchnology and Invstmnt, 2017, 8, ISSN Onln: ISSN Prnt: Invstng on th CAPM Prcng Error José Carlos d Souza Santos, Elas Cavalcant Flho Economcs Dpartmnt, Unvrsdad d São Paulo, São Paulo, Brazl How to ct ths papr: d Souza Santos, J.C. and Flho, E.C. (2017) Invstng on th CAPM Prcng Error. Tchnology and Invstmnt, 8, Rcvd: January 4, 2017 Accptd: Fbruary 19, 2017 Publshd: Fbruary 22, 2017 Copyrght 2017 by authors and Scntfc Rsarch Publshng Inc. Ths work s lcnsd undr th Cratv Commons Attrbuton Intrnatonal Lcns (CC BY 4.0). Opn Accss Abstract W tstd an nvstmnt stratgy basd on th prcng rror of th CAPM modl. Startng wth th Markowtz (1952) [1] mthodology, w rplacd th standard xpctd rturns vctor wth th xpctd rrors vctor from th CAPM modl, assumng that such rrors ar nonzro and prsst ovr tm. Whn valuatd ovr th ntr xamnd prod, all of th rsultng portfolos outprformd th markt portfolo. Excpt for som shortr prods, our hypothss was fully confrmd. That s, th prformanc of our alpha portfolos was sgnfcantly bttr than th markt portfolo. In othr words, th prcng rror of th CAPM modl sms to b nonzro and to hav an nrtal componnt. Kywords CAPM, Portfolo Optmzaton, EWMA, Sharp Rato, Crtanty Equvalnc 1. Introducton Th goal of ths study s to dtrmn whthr th prcng rror of th CAPM modl by Wllam Sharp (1964) [2] and Lntnr (1965) [3] xhbts an nrtal componnt. Fama and Frnch (2004) [4] stat that CAPM s stll wdly usd for prcng n both acadma and ndustry. Thr s, howvr, a vast ltratur, ncludng Banz (1981) [5], Rosnbrg, Rd and Lanstn (1985) [6], Fama and Frnch (1993) [7], Jgadsh and Ttman (1993) [8] and Fama and Frnch (2014) [9], whch shows that th modl s asst prcng s flawd. Th CAPM modl stats that th xpctd rturn of a gvn asst satsfs th followng formula: E R R = α + β E R R ( f ) ( m f ) whr R s th rturn on asst ; R s th rsk-fr rturn; f R m s th markt rturn; and α s th xpctd prcng rror of asst. Not, howvr, that th modl also assums that th prcng rror s zro, that s, that α = 0, { 1, 2,, N}, whr N s th numbr of avalabl assts. DOI: /t Fbruary 22, 2017

2 Ths papr, n contrast, assums that th rror s non-zro, and tsts whthr ths prcng rror s prsstnt and may b proftably tradd on. Spcfcally, w tst whthr xcss rturns may b obtand by takng long and short postons n assts accordng to th proportons gvn by th modl s bta paramtrs, so that th portfolo rturns rflct th modl prcng rror. Our motvaton s as follows. Both th fnancal markt as a whol and ndvdual nvstors gnrally assss th prformanc of a stock portfolo smply by comparng th portfolo rturn to that of a stock ndx. Th portfolo s thrfor valuatd solly by ts markt rsk, dsrgardng othr rsk factors such as thos spcfd n Fama and Frnch (1993) [7]. W constructd optmal portfolos usng th Markowtz (1952) [1] mthodology, but rplacng th xpctd rturns vctor and th varanc and covaranc matrcs wth th xpctd prcng rror and ts rspctv varanc and covaranc matrcs. W thn compard th rturn of th rsultng port- folos wth th rturn of th markt portfolo. Th rsult was postv nsofar as th constructd ( alpha ) portfolos yldd gratr-than-markt rturns whn consdrd ovr th ntr prod of avalabl data. On th othr hand, ovr som shortr prods, th prformanc of th alpha portfolos, as masurd va Sharp ratos and crtan quvalnts, dd not sgnfcantly mprov on that of th markt portfolo. 2. Ltratur Rvw Th cor of Modrn Portfolo Thory s basd on th work of Markowtz (1952) [1], whch arnd hm th 1990 Nobl Prz n Economcs. Ths cor corrsponds to th st of Man-Varanc (MV) typ modls. Th MV modls analys th bhavor of a rsk-avrs nvstor ovr a fnt tm horzon. Startng from a gvn ntal allocaton th nvstor chooss a portfolo by slctng assts and thr rspctv quantts from a st of N dstnct avalabl assts. Ths dcson s mad basd on th nvstor s knowldg of 1) th xpctd rturns and 2) th varancs and covarancs of th avalabl assts. Th nvstor bulds th portfolo so as to maxmz xpctd rturns for a gvn lvl of rsk. Whl th basc nsght of th MV modls s rlatvly smpl, thr mplmntaton prsnts two srous problms. Frst, as rportd by D Mgul, Garlapp and Uppal (2007) [10], t s dffcult to obtan accurat stmats of th xpctd rturns vctor as wll as of th varanc and covaranc matrcs of th optmal portfolos. Scond, th ltratur uss th captal asst prcng modl (CAPM) dvlopd by Wllam Sharp (1964) [2] and John Lntnr (1965) [3] to xplan th xpctd asst rturns. CAPM rls on th assumpton that thr s a lnar rlatonshp btwn th markt factor, rprsntd by th xcss rturn of th markt portfolo, and th snstvty of th asst to that markt factor, known as th asst bta. Howvr, whl th CAPM modl s wdly usd both n acadma and n n- 68

3 dustry (s Fama and Frnch (2004) [4]), svral studs show that t substantally fals to xplan asst rturns. Spcfcally, CAPM consstntly ylds nonzro prcng rrors. Th aformntond studs nclud for nstanc 1) Fama and Frnch (1993) [7], who rport that th CAPM s unabl to xplan som rturns whch ar rlatd to th book-to-markt rato and th markt valu (thy morovr propos a nw, multfactor CAPM modl to fx ths problm); and 2) Jgadsh and Ttman (1993) [8], who show that asst rturns ar postvly rlatd to past prformanc and thrfor cannot b xpland solly by th asst bta and th markt prmum. Hnc th CAPM, dspt ts wdsprad us, dsplays som srous mprcal falurs. It thrfor mght b possbl to buld portfolos whos postons tak advantag of th modl s prcng falurs. Spcfcally, w nvstgat whthr t s possbl to arn xcss rturns by applyng th mthodology proposd by Markowtz (1952) [1], but rplacng th xpctd rturns vctor wth th CAPM s xpctd prcng rror vctor. 3. Data In ths papr w usd BOVESPA (Bolsa d Valors d São Paulo-Brazl) stock prcs, natonal Brazlan rsk-fr ntrst rats, and a srs of thortcal portfolo rturns corrspondng to actual markt bhavor durng th 02/01/2001 to 02/01/2015 prod. Both th markt rturn hstory and th rsk-fr rat wr obtand from NEFIN 1, whl th asst prcs wr collctd from Economatca 2. Th markt rturn was dfnd as th xcss rturn (.., rturn mnus th rsk-fr rat) of a thortcal markt portfolo dvlopd by NEFIN. Th rsk fr rat was gvn by th 30-day DI swap rat. Assts had to mt th followng crtra to b ncludd n th markt portfolo: a) B th most tradd asst of th company, that s, th on wth th hghst tradng volum durng th prvous yar; b) Hav bn tradd durng at last 80% of th days of th prvous yar, wth a man fnancal volum gratr than R$500, pr day. If th asst was frst lstd n th prvous yar, th prod bgns on th lstng day and nds on th last day of th yar; c) Th asst must hav bn lstd bfor Dcmbr of th prvous yar; Th asst prc hstory usd th sam crtra appld n th slcton of th assts n th markt portfolo, that s, on a gvn day th rturn of an asst was usd only f on that day th asst blongd to th markt portfolo. Morovr, f an asst was not tradd n a gvn day thn w assumd that ts prc rmand constant at th last obsrvd prc. 4. Mthodology In ths scton w prsnt th mthodology appld n ths work. Frst, w pr- 1 Brazlan Cntr for Rsarch n Fnancal Economcs of th Unvrsty of São Paulo (Núclo d Psqusas m Economa Fnancra, 2 A platform whch provds nformaton about Latn Amrca s stock markts, govrnmnt bonds, th fund ndustry and varous ndcators ( 69

4 snt how th portfolos wr buld. Aftr that w show th prformanc masur usd to compar th alphas portfolos wth th bnchmark usd, th markt rturn Mthod for Alpha Portfolo Constructon In ordr to buld th alpha portfolos, w follow som stps. Frst, w slct a tm wndow, thn, n ths tm wndow, w stmat th CAPM rror for ach asst n ach prod of tm. Followng, w forcast th xpctd alpha for th nxt prod and w us ths xpctd rror as th xpctd rturn n a Markowtz portfolo framwork. W apply som constrans on th portfolos wghts, as consqunc w hav dffrnt sorts of portfolos, ach on dfnd undr dffrnt wghts constrants. W also apply a stop mchansm n ordr to lmt losss. In th nxt subsctons w xpland n dtal how w prform th alphas stmaton, th alpha forcast and how w dfn th portfolos wghts Alpha Vctor and Varanc and Covaranc Alpha Matrcs Th man workng hypothss of ths study s that th CAPM modl xhbts nrtal prcng rrors. That s, w assum that f th modl undrprcs (ovrprcs) an asst durng a gvn prod, thn t tnds to do lkws durng th subsqunt prod. Gvn ths assumpton, w shall chck whthr t s possbl to xplot ths bhavor to arn xcss rturns. Invstors can tak advantag of th prcng rror of a gvn asst to th xtnt that ths rror s prdctabl. If th modl s undrvalung th asst and th nvstor s awar of ths, sh should tak a long poston n th asst and smultanously short a markt poston whos wght corrsponds to th systmc rsk of th asst, that s, to th β of th CAPM. W call ths stratgy nvstng n th assts alpha. Basd on ths stratgy ths papr studs portfolos that wr constructd usng not th calculatd xpctd rturn and rsk, but nstad th xpctd forcast rror and ts corrspondng varanc and covaranc matrcs. In ths scton w xplan frst th stmaton of th asst rror and thn th stmaton of th varanc and covaranc matrcs Alpha Vctors To obtan th alpha stmatd vctor for a spcfc day h t was ncssary to stmat th CAPM modl for ach asst gvn th nformaton avalabl at that tm, so as to yld th xpctd stmaton rror of th modl. Consdr th CAPM modl E R = α + β E R R = α + β E R (1) wth ( ) t t ( ) t t ( ) t t m f t m Cov t ( R, Rm) β Et ( β t ) Vart ( Rm) ( ) ( ) ˆ ( ) ( ˆ ) = (2) α E α = E R β E R = E R β R (3) t t t t t m t t m 70

5 whr R s th xcss rturn of asst ; α t s th xpctd prcng rror of asst condtonal on th nformaton avalabl n prod t ; β t s th systmc rsk of asst condtonal on prod t ; R m s th markt rturn; and R s th rsk-fr rturn. f Not that n ordr to stmat th xpctd prcng rror α t of th CAPM modl w must frst comput ach of ts ralzd past valus. Blow w frst summarz th procss nvolvd, and latr dtal ts stps. W prform th followng procdur to stmat th xpctd forcast rror for a subsqunt prod. Gvn ach asst { 1, 2,, N} and ach prod h { T + 1, T + 2, }, whr N s th numbr of avalabl assts; H th st of days wth avalabl data; and T th numbr of days usd n th modl stmaton: a) St th valu of T, th numbr of days usd n th modl stmaton; b) Estmat β ˆ h ; c) Calculat ˆ ˆ α = R, t, β Rtm,, t ( hh, 1, h 2,, h T+ 1 th h ),.., calculat ach modl rror n th stmaton sampl; d) Gvn th squnc ( ˆ α, ˆ α,, ˆ α hh, h 1, h h T+ 1, h), whch was stmatd n th prvous stp, stmat α ˆ h, th xpctd valu of α ˆ h ; W now xplan th dtals nvolvd n th calculatons abov Calculatng β To obtan β ˆ h frst dfn Rt, R h Σ = R th, t, R R h tm, R (4) mh Rtm, R mh whr R t s th sampl man for asst durng th stmaton prod, whch covrs from ( h T + 1) to h. Thrfor Σ th, s th cross product of th dvaton from th man of th xcss rturn of asst at tm t. Hnc h( Rt, ) ( ) ( ) R Var E (, ) = = t h t, Σ Σ Var, h h th h R tm, Cov h Rt,, Rtm, Varh R tm, so that Σ h s th varanc-covaranc matrx condtonal on h of th xcss rturns vctor of asst. W now stmat Σ h assumng that ts valu may b obtand va th EWMA (xponntally wghtd movng avrag) of th squnc ( Σ, Σ,, Σ h T, h h T+ 1, h hh, ) : ( 1 ) ( 1 ) ( ) Σ ˆ = Σ + Σ + Σ Σ 2 T δ δ δ δ δ δ δ h hh, h 1, h h 2, h h T, h whr ˆΣ h stmats Σ h and δ was fxd at W fnally hav that Cov h ( R, Rm) Var ( ) h Rm (5) ˆ β = (7) h 71

6 Calculatng α ˆ, 1, 2,, N h Gvn β { } and h { T 1, T 2,, H} α th, = R ˆ β R, t, h tm, + + w can obtan { } { } {, 1,, 1} 1, 2,, N, h T + 1, T + 2,, H and t hh h T+ (8) Analogously to what was don n th stmat of Σ h w us EWMA to stmat α ˆ h as follows: ( 1 ) ( 1 ) ( 1 ) 2 T ˆ α = λαˆ + λ λ ˆ α + λ λ ˆ α + + λ λ ˆ α (9) h hh, h 1, h h 2, h h T, h whr α ˆ h stmats α h and th valu of λ was chosn to mnmz th stmat s man quadratc rror Alpha Varanc and Covaranc Matrcs W now xplan th stps nvolvd n stmatng th varanc and covaranc matrcs of th alpha assts at ach momnt n tm. W ntnd to stmat: Ω h ( α ) Cov( αα ) Var 1 1 N = Eh (10) Cov( αnα1 ) Var ( αn ) Usng th valus of α ˆt h whch wr stmatd n th prvous scton w hav that: ˆ α ˆ α 1th 1h Ω ˆ = ( ˆ α ˆ α ˆ α ˆ α 1 1 ), th th h Nth Nh ˆ α ˆ α Nt h N h t h T h {,, } whr ˆ α 1 h s th sampl man of ˆt h t hh, 1,, h T+ 1. Now, as n th prvous scton, w mploy EWMA to obtan th followng stmat: α, { } ( ) ( ) ( ) 2 T h δ δ δ δ δ δ δ hh h 1h h 2h h Th (11) Ω ˆ = Ω + 1 Ω + 1 Ω Ω (12) whr δ = 0.96 s th sam dcay xponnt whch w usd to stmat ˆΣ h Portfolos Lt us now xplan th ruls usd to buld th portfolos. W mployd thr mthods to assmbl th portfolos: th frst on, calld Naïv, s th smplst of th thr; th othr two, whch w call Markowtz and Long-only Markowtz, ar basd on th Markowtz (1952) [1] mthodology of optmal portfolos. Th lattr two dffr from ach othr n trms of th rstrctons mposd on ach of thm. For ach momnt h { T 1, T 2,, H} + + w computd th portfolos of all thr mthodologs usng th valus obtand as dscrbd n scton Th Naïv portfolos Th wgths of ach assts alpha n th Nav portfolos ar basd solly on th 72

7 sgns of th xpctd alphas. Th procdur nvolvs thr stps: 1) th assts wr classfd accordng to th sgns of thr xpctd alphas; 2) assts wth postv alphas wr assgnd a portfolo wght of 2 N ( + ), whr N ( + ) s th total numbr of assts wth postv alphas; and 3) assts wth ngatv alphas wr assgnd a portfolo wght of 2 N( ), whr N( ) s th total numbr of assts wth ngatv alphas. Ths thrby producd portfolos wth (quallywghtd) long postons n assts wth postv alphas and (qually-wghtd) short postons n assts wth ngatv alphas n vry prod Th Markowtz Portfolos Th Markowtz portfolos wr dsgnd to maxmz th xpctd rturn condtonal on a spcfc lvl of rsk. Th rsk lvl mtrc usd was th standard dvaton of th markt portfolo obsrvd n th last 252 days rlatv to th rfrnc dat ( σ markt h ). Th Markowtz portfolos wr thrfor obtand by solvng th followng optmzaton problm: max w α wt st.. t hh w tι = 1; w tω twt σ ; markt h w w w ; mn t, max whr w t s th (1 N ) wght vctor; ι s an ( N 1) vctor consstng of 1s; w mn s th mnmum accptabl valu of w t ; w max s th maxmum accptabl valu of wt ; α hh s th ( N 1) xpctd alphas vctor computd prvously; and Ω t s th ( N N ) alpha varanc and covaranc matrx Th Long-Only Markowtz Portfolos Th long-only Markowtz portfolos dffr from th plan Markowtz portfolos n that th formr hav an addtonal rstrcton on th portfolo wghts: vry wght must b postv, so that thr ar no short postons n ndvdual assts. In th long-only cas th wghts thrfor follow from th soluton of th followng optmzaton problm: max w α wt st.. t w tι = 1; w tω twt σ ; markt h w 0 t, Bounds on Lvrag Th short postons allowd by th Markowtz portfolos mply th possblty of lvrag, n th sns that th sum of th wghts of long postons of th portfolo may xcd 100%. W mpos a lowr bound w mn on th wghts to lmt th lvrag of th portfolo. Th goal was thrfor to fnd a valu for w mn that solvd th fol- hh (13) (14) 73

8 lowng problm: max wmn = 1 st.. N N w N w L; w = 1; w w ; t, max = 1 = 1 (15) whr L s th maxmum lvrag. Hr w us th sum oprator (ovr all assts) nstad of th vctor ι to hghlght that th total numbr of avalabl assts affcts th soluton. W llustrat th ssu wth an xampl. Assum that w hav 12 avalabl assts and that w max = 2 and L = 10. If w mn = 1 thn th maxmum lvrag vctor ( w max L ) 3 would b gvn by: w = ( 1, 1, 1, 1, 1, 1, 1, 1, + 1, + 2, + 2, + 2, + 2) L max Not that all of th rstrctons ar satsfd xcpt for th lvrag bound, snc th sum of th absolut valus of th wghts s qual to 13. Ths mpls that th valu of w mn must b st hghr: n ths xampl t must b qual to 0.5, as xpland blow. w = ( 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, + 1.5, + 2, + 2) L max W us th followng numrcal procdur to fnd th rqurd valu of w mn. Gvn valus for w max, N, and L w obtan th vctor of maxmum lvrag for a tral w mn by rpatdly (1) chckng whthr th rstrctons wr satsfd and thn (2) changng th valus of w mn untl all of th rstrctons wr satsfd. In ths study w chos to spcfy a rasonabl valu for L and at th sam tm pck a valu for w max that s hgh nough that t dos not consttut an actv rstrcton for th soluton of th problm Stoploss W usd a stoploss mchansm: th portfolo poston was st to zro whnvr th poston accumulatd a loss gratr than or qual to 25% Alpha Portfolos By followng th prvous stps, w hav so far assmbld a portfolo for ach day of th prod, xcpt for th ntal T arly days, whch wr usd only for th ntal stmats. Thus, on can consoldat th 21 portfolos nto a fnal portfolo as follows: a) usng th wghts obtand wth th portfolo assmbld wth th nformaton avalabl at th tm T, w nvst on dat T + 1; b) on dat T + 2 w accumulat th rturn obtand n th prvous stp and assmbl a nw portfolo usng th nformaton avalabl on T + 1; d) w contnu th procdur untl w obtan 21 portfolos; ) on dat T w st to zro th poston of th portfolo that w hav obtand durng th prvous 21 days and nvst ts cumulatv valu on a port- 74

9 folo assmbld usng th nformaton avalabl up to dat T ; f) w contnu th procdur by vry day dscardng a portfolo and rplacng t wth a nw on, so that w ar always nvstng n a combnaton of 21 portfolos. Th nd rsult s a portfolo consstng always of svral portfolos wth dstnct maturts. Th frst portfolo was assmbld usng only th ntal nformaton and s nvstd n for only on day, whl th oldst portfolo blongs to th portfolo bgnnng 21 days ago and wll b dscardd th followng day. Th prformanc of th alpha portfolo s gvn by th avrag prformanc of th 21 portfolos that consttut t. W choos to analys th portfolos n ths way, so our rsults wr lss snstv to th data that th analyss bgns Constant Wghts or Quantts W dal wth varous typs of portfolos. Each alpha s a portfolo consstng of th asst and th markt, whch bcoms mor complx as w add mor and mor alpha assts. Ths n turn bcom vn mor laborat whn w consoldat portfolos of dstnct maturts nto a sngl portfolo. As w accumulat th portfolo rturns, ths procss rass th followng quston: what should b kpt constant ovr tm, th asst wghts or th asst quantts? W consdrd both approachs. W frst kpt wghts constant w call ths th Equal Wght approach. Rgardlss of past prformanc all wghts wr kpt fxd and qual to th orgnally calculatd wghts. Ths nvolvs of cours sllng th outprformng assts and buyng th undrprformng assts so that so that thy rman constant as portons of th whol portfolo. In th scond approach, calld Valu Wght, th asst wghts vary dpndng on thr past prformanc. Outprformng assts gan wght and undrprformng assts los wght n th portfolo. Ths s quvalnt to kpng constant th ntally purchasd quantts: no rbalancng trads ar mad Prformanc Mtrcs Analyss In ths scton w dscuss th mtrcs usd to assss portfolo prformanc. As xpland prvously, our mthodology ylds varous portfolo typs, charactrzd by thr rstrctons on th maxmum and mnmum wghts, th maxmum amount of lvrag, and othr crtra. W apply a st of tsts to compar th prformanc of ach portfolo wth that of th markt portfolo. Th mtrcs usd n ths comparsons ar basd on D Mgul, Garlapp and Uppal (2007) [10] and xpland blow Sharp Rato Tst W bgn by computng th Sharp rato of ach of th portfolos. For ach { 1, 2,, N} w hav: µ ι SRι = (16) σι whr µ ι, σ ι and SR ar rspctvly th man rturn, th standard dva- 75

10 ton, and th Sharp rato of portfolo. W thn us an approach suggstd by Jobson and Kork (1981) [11] and amndd by Mmml (2003) [12] to tst whthr th dffrncs among th Sharp ratos ar sgnfcant. Lt whr { 1, 2,, } H : 0 0 SR j SR M = (17) j N M and M s th markt portfolo. W thn comput th z ˆM statstc, whos asymptotc dstrbuton s th standard normal dstrbuton 4 : such that ˆ z M ˆ σµ ˆ ˆ ˆ σ Mµ M = (18) ˆ υ ˆ µµ ˆ ˆ υ= 2 ˆ σ ˆ σ 2 ˆ σσˆ ˆ σ + ˆ µσ ˆ + ˆ µ ˆ σ σ T M 2 M M IM M M M 2 2 ˆ σσˆ M (19) Thrfor, just as n a t tst, w nd only compar th valu obtand abov wth th crtcal valu gvn by th normal dstrbuton: f th crtcal valu s lowr than th computd valu thn w rjct th null hypothss that th Sharp ratos of th two portfolos ar qual Crtanty Equvalnc Analyss Th crtanty quvalnt tst gos largly as abov. It also conssts n calculatng a prformanc mtrc for all portfolos followd by a dffrnc tst btwn th assmbld portfolos and th markt portfolo. In ths nw cas th mtrc s th crtanty quvalnt 3 of th portfolo, obtand as follows: γ 2 EQ = ˆ µ ˆ σ (20) 2 whr { 1, 2,, N} and γ s th rsk avrson paramtr. Havng frst computd th crtanty quvalnt for all portfolos usng (20), w prform a dffrnc tst on th valus so obtand. γ 2 γ 2 Spcfcally, lttng fm ( v) = µ σ µ M σm 2 2, w tst th null hypothss H : 0 f ( v ) = 0, knowng that: whr T ˆ f f Ψ ( ( ˆ M = T fm v) fm ( v) ) ~ N 0,? Θ v v 2 σ 0 0 σ M 2 σ 0 0 M σm Θ= σ 2σ M σM 2σM (21) (22) 3 Th crtanty quvalnt s th ncras n th rsk-fr rat that would rndr th agnt ndffrnt btwn th rsky nvstmnt and th rsk-fr nvstmnt. As was pontd out by D Mgul, Garlapp and Uppal (2007) [10], Equaton (20) corrsponds not to th crtanty quvalnt but nstad to th xpctd utlty of a man-varanc nvstor, whch s n turn approxmatd by th crtanty quvalnt of a quadratc-utlty nvstor. 76

11 W analys th rsults as bfor: f th computd valu s hghr than th crtcal valu thn w rjct th null hypothss. Not that w ar sttng th rsk avrson paramtr γ qual to Rsults W now dscuss th prformanc of th portfolos. Th analyss s organzd as follows: w frst compar th rturns of th alpha portfolos wth th rturn of th markt portfolo ovr th whol prod; w thn compar th alpha and markt portfolos by usng th prformanc mtrcs dscrbd n scton 4.2. Th varous typs of portfolos nvolvd n ths papr ar summarzd n Tabl 1, whch spcfs for ach portfolo th wghtng convnton (qual wght or valu wght), th abbrvatd nams, and som xplanatory rmarks. Th frst portfolo s th markt portfolo, whos prformanc s usd as th bnchmark. Th portfolos ar classfd nto four typs, ach contanng two portfolos: on wth qual wghts and on wth valu wghts as xpland n scton Th frst of th four typs ar th Markowtz long-only portfolos, whch consst of portfolos wthout any short postons, as xpland n scton Scond, th nav portfolos follow th smplr procdur dscrbd n scton Thrd, thr ar th (rstrctd-wght) Markowtz portfolos whos mthodology has bn dscrbd n scton and whos wghts ar rstrctd to a spcfc rang, dsplayd n Tabl 1 as gong from X to Y. Fnally, th boundd lvrag portfolos, whch wr dscrbd n scton , hav lvrag lmtd to Alpha Portfolo Analyss W now compar th rturns of th alpha and markt portfolos. Tabl 2 dsplays th cumulatv avrag rturns for ach portfolo and th markt portfolo Tabl 1. Analysd portfolos. Typ Wght Abbrvaton Rmark Markt - Mrkt - Markowtz long-only Equal Valu m-long.e m-long.v - Markowtz lvrag Equal Valu m-lvrg.e m-lvrg.v Maxmum lvrag qual to 10 Naïv Equal Valu nav.e nav.v - Markowtz Equal Valu m-x.y.e m-x.y.v Wghts must b btwn X and +Y Th frst column dsplays th nam of th portfolo, th scond column ndcats whthr th portfolo was assmbld wth constant wghts (Equal) or wghts varyng accordng to past prformanc (Valu), and th thrd column ncluds som xplanatory rmarks. 77

12 Tabl 2. Man rturn rats. Man rturns (% pr day) Prod Portfolos Full prod Mrkt m-long.e m-long.v m-lvrg.e m-lvrg.v nav.e nav.v m e m v m e m v m e m v m e m v Avrag pr-day (% p.d.) rats of cumulatv rturns of th alpha and markt portfolos. Th rats ar groupd nto two-yar ntrvals. Th Full prod column dsplays th avrag pr-day rats ovr th ntr prod. n trms of daly rturn (% p. d.) and th valus wr groupd nto two-yar ntrvals. Not that th rturns of th alpha portfolos ar oftn much hghr than thos of th markt portfolo. Ths s not tru, howvr, btwn 2004 and 2009, whn alpha rturns gnrally t wth or ar lowr than th markt rturns. Th rsults ndcat a partal succss n assmblng portfolos that bat th markt. Though th alpha rturns ar hghr ovr th whol prod, ths dos not occur consstntly ovr tm: thr ar farly long prods durng whch th stratgs do not accuratly prdct th asst alphas. It follows that nrtal bhavor of th asst alphas whch was usd as a hypothss n ths work cannot b fully confrmd. On th othr hand, ths nrta dd gnrat sgnfcant gans whn th ntr prod s consdrd. As w shall s shortly, th rsults do show that th rsultng gans justfy th addtonal rsk ncurrd. W now chck ths usng th mtrcs whch wr dscussd n sctons 2.3 and Sharp Rato Analyss W now turn our attnton to comparng th Sharp rato of th portfolos. Th rsults ar summd up n Tabl 3. Th frst column nams th portfolos, th 78

13 Tabl 3. Sharp rato dffrnc tsts. Portfolo Sharp rato ˆ ˆ µ σ Statstc ˆ ~ ( 0,1) z N p-valu Mrkt m-long.e m-long.v m-lvrg.e 0.059* m-lvrg.v 0.059* nav.e nav.v m e m v m e 0.061* m v 0.059* m e m v m e m v Gvn ach portfolo w tstd th hypothss H : ˆ µ ˆ σ ˆ µ 0 M ˆ σ M = 0, whr µ s th man portfolo rturn, σ s ts standard dvaton, s th ndx of th alpha portfolo and M s th markt portfolo. Sgnfcanc lvls: ***1%, **5%, *10% M scond column lsts thr Sharp ratos, and th thrd column shows th z ˆM statstc rgardng th dffrnc tst btwn th Sharp ratos of th alpha and markt portfolos that was dscussd n scton Fnally, th last column rports th tst p-valu. As shown n th Tabl, th markt portfolo had a Sharp rato of 0.015, th lowst among all analysd portfolos. Hr ar th rmanng portfolos n dcrasng ordr of prformanc as masurd by th Sharp rato: m e, m v, m-lvrg.e, m-lvrg.v m e, m e, m v, m e, m v, m v, nav.e, nav.v, m-long.v and m-long.e. Thr Sharp ratos rangd from to Although all of th alpha portfolos hav hghr Sharp ratos than th markt portfolo, only th top four wr sgnfcantly hghr, all four at th 10% sgnfcanc lvl. Not also that thr was lttl dffrnc n prformanc btwn th qual-wghtd and valu-wghtd portfolos. As on can s th prformanc was ssntally th sam rgardlss of wght usd Crtanty Equvalnt Analyss W now analys th alpha portfolos va th crtanty quvalnt dffrncs tsts dscussd n scton As prvously xpland, ths tsts compar th crtanty quvalnt of ach alpha portfolo wth th markt portfolo. W st out th rsults n Tabl 4, whos columns dsplay th followng nformaton from lft to rght: abbrvatd portfolo nam, valu of th crtanty quvalnt, valu of th Ψ M statstc, and th tst p-valu. 79

14 Tabl 4. Crtanty quvalnt dffrnc tsts. Portfolo 1 Crtanty quvalnt ˆ µ ˆ σ K 2 Statstc Ψ ~ ( 0,1) 2 K M N p-valu Mrkt m-long.e m-long.v m-lvrg.e ** m-lvrg.v ** nav.e nav.v m e * m v m e ** m v ** m e ** m v ** m e ** m v * Gvn ach portfolo w tstd th hypothss H : Ψˆ = T 0 ( f ( vˆ ) f ( v) ) = 0, whr M M M γ 2 γ 2 fm ( v) = µ σ µ σ M M 2 2, µ s th man portfolo rturn, σ s th standard dvaton, s th ndx of th alpha portfolo, M s th markt portfolo, and γ s th rsk avrson paramtr, st at γ = 1. Sgnfcanc lvls: ***1%, **5%, *10%. As n th prvous tst, th markt portfolo had th worst rsult among all portfolos. Th valus for th alpha portfolos, howvr, ar not always statstcally hghr than th markt portfolo. Hr ar th rmanng portfolos n dcrasng ordr of prformanc as masurd by th crtanty quvalnt: m e, m e, m e, m v, m v, m-10:30.v, m-lvrg.e, m-lvrg.v, m e, m v, nav.e, nav.v, m-long.v and m-long.e. Nn of th top alpha portfolos valus hav crtanty quvalnts statstcally abov th markt portfolo at a 10% sgnfcanc lvl. Th crtanty quvalnt tst rsults ar n gnral vry smlar to th Sharp rato tst rsults. Th lsts of portfolos ordrd by dcrasng prformanc ar smlar for th dffrnt mtrcs usd. Thr s morovr lttl dffrnc n prformanc, agan as masurd by th varous mtrcs, btwn qually-wghtd and valu-wghtd portfolos of othrws th sam typs. 6. Conclusons W valuatd an nvstmnt stratgy by bttng on prcng rrors mad by th CAPM modl, undr th hypothss that ths rrors ar non-zro and prsst ovr tm. In ordr to do so, w frst computd for ach asst ts xpctd prcng 80

15 rror (th asst alpha), dfnd as th xpctd rturn whn on taks a long poston n th asst whl shortng th markt wth wght quvalnt to th asst rsk (bta). W ssntally bult optmal Markowtz (1952) [1] portfolos, rplacng th xpctd rturns vctor by th xpctd rror vctor. Our rsults show that ovr th ntr prod of analyss, all of th alpha portfolos obtand abov-markt rturns. Ths rsults wr not sgnfcantly affctd by th choc of qual-valud or valu-wghtd portfolos. Excpt ovr som shortr prods, our hypothss was fully confrmd, that s, th prformanc of our alpha portfolos was sgnfcantly bttr than that of th markt portfolo. Furthrmor, two mtrcs wr usd to assss portfolo prformanc, th Sharp rato and th crtanty quvalnt. Both mtrcs confrmd that th prformancs of th alpha portfolos wr bttr than that was obsrvd for th markt portfolo. In othr words, th prcng rror of th CAPM modl has an nrtal componnt that allows on to obtan rturns that ar sgnfcantly abov markt rturns. Rfrncs [1] Markowtz, H. (1952) Portfolo Slcton. Th Journal of Fnanc, 7, [2] Sharp, W.F. (1964) Captal Asst Prcs: A Thory of Markt Equlbrum undr Condtons of Rsk. Th Journal of Fnanc, 19, [3] Lntnr, J. (1965) Th Valuaton of Rsk Assts and th Slcton of Rsky Invstmnts n Stock Portfolos and Captal Budgts. Th Rvw of Economcs and Statstcs, 17, [4] Fama, E.F. and Frnch, K.R. (2004) Th Captal Asst Prcng Modl: Thory and Evdnc. Journal of Economc Prspctvs, 18, [5] Banz, R.W. (1981) Th Rlatonshp btwn Rturn and Markt Valu of Common Stocks. Journal of Fnancal Economcs, 9, [6] Rosnbrg, B., Rd, K. and Lanstn, R. (1985) Prsuasv Evdnc of Markt Inffcncy. Th Journal of Portfolo Managmnt, Insttutonal Invstor Journals, 11, [7] Fama, E.F. and Frnch, K.R. (1993) Common Rsk Factors n th Rturns on Stocks and Bonds. Journal of Fnancal Economcs, 33, [8] Jgadsh, N. and Ttman, S. (1993) Rturns to Buyng Wnnrs and Sllng Losrs: Implcatons for Stock Markt Effcncy. Th Journal of Fnanc, 48, [9] Fama, E.F. and Frnch, K.R. (2014) A Fv-Factor Asst Prcng Modl. Journal of Fnancal Economcs, 116, [10] Dmgul, V., t al. (2007) Optmal vrsus Nav Dvrsfcaton: How Inffcnt Is th 1/N Portfolo Stratgy? Th Rvw of Fnancal Studs, 22, [11] Jobson, J.D. and Kork, B.M. (1981) Prformanc Hypothss Tstng wth th Sharp and Trynor Masurs. Th Journal of Fnanc, 36,

16 [12] Mmml, C. (2003) Prformanc Hypothss Tstng wth th Sharp Rato. Fnanc Lttrs, 1, Submt or rcommnd nxt manuscrpt to SCIRP and w wll provd bst srvc for you: Accptng pr-submsson nqurs through Emal, Facbook, LnkdIn, Twttr, tc. A wd slcton of journals (nclusv of 9 subjcts, mor than 200 journals) Provdng 24-hour hgh-qualty srvc Usr-frndly onln submsson systm Far and swft pr-rvw systm Effcnt typsttng and proofradng procdur Dsplay of th rsult of downloads and vsts, as wll as th numbr of ctd artcls Maxmum dssmnaton of your rsarch work Submt your manuscrpt at: Or contact t@scrp.org 82