Modern Portfolio Theory (MPT) Statistics

Transcription

1 Modrn Portfolo Thory (MPT) Statstcs Mornngstar Mthodology Papr Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or n part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd.

2 Introducton Modrn Portfolo Thory Statstcs (MPT statstcs) ar asd on th Captal Asst Prcng Modl (CAPM) of xpctd rturns dvlopd y Nol laurat Wllam Sharp and othrs n th arly 1960s. Th CAPM s asd on Modrn Portfolo Thory (MPT) dvlopd n th 1950s y Sharp s tachr and co-laurat Harry Markowtz. In th trmnology of anothr Nol laurat, th lat Mlton Frdman, MPT s a normatv thory, manng that t s a prscrpton for how nvstors ought to hav. In contrast, th CAPM s a postv thory n that t mant to a dscrpton of how nvstors do hav. Th CAPM s asd on MPT n that t assums th all nvstors follow th prscrptons of MPT. Th CAPM sparats th xcss rturn (.., total rturn mnus th rturn on a rsk-fr scurty) of ach scurty nto two componnts: systmatc xcss rturn and unsystmatc (or dosyncratc) rturn. Systmatc xcss rturn s drctly proportonal to th xcss rturn of th markt portfolo. Th rato of th xcss rturn of th markt to th systmatc xcss rturn of th scurty n quston s th scurty s ta. Bta masurs how snstv th xcss rturn on a scurty s to th xcss rturn of th markt as a whol. On of th man mplcatons of th CAPM s that th xpctd xcss rturn on a scurty s drctly proportonal to systmatc rsk as masurd y ta and s not rlatd to any othr varal. Ths mans that thr ar no rwards for takng on unsystmatc rsk. In an ffcnt markt n whch th CAPM holds, th only way to otan an xpctd rturn aov that of th markt portfolo s to tak on a ta aov on. In th 1970 s, Mchal Jnsn proposd a prformanc masur for actvly managd funds that s asd on th CAPM calld Jnsn s alpha or smply alpha. Th da of alpha s that a managr should not rcv crdt for achvng aov-markt prformanc y takng on systmatc rsk as masurd y ta. Alpha s th avrag xcss rturn that a portfolo achvs aov and yond that could hav n otand from poston n th markt portfolo, lvrd or d-lvrd so as th hav th sam ta as th fund. Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd.

3 Introducton (contnud) Strctly spakng, th CAPM cannot appld n th ral world caus rturns on th markt portfolo ar unosrval. To mak alpha and ta practcal masurs at frst, road stock markt ndxs wr usd as proxs for th markt portfolo. As funds cam mor spcalzd, mor narrow nchmarks wr dvlopd to track rturns on th mor narrowly dfnd sourcs of systmatc rsk and rturn. Today t s common practc to masur alpha and ta usng a narrowly dfnd nchmark that s chosn to rprsnt th man sourc of th systmatc rsk of th fund ng analyzd. In addton to alpha and ta, a thrd MPT statstc s R-squard. R-squard masurs th strngth of th rlatonshp twn xcss rturns on th nchmark and xcss rturns on th fund ng analyzd. MPT statstcs ar calculatd from a comparson of a fund s xcss rturns and th nchmark s xcss rturns. Unlss a tm horzon s spcfd, Mornngstar s MPT statstcs ar asd on thr yars of monthly rturns. Mornngstar calculats MPT statstcs for ach fund, usng a standard st of nchmarks for ach asst group. Mornngstar also calculats st-ft MPT statstcs, whch ar asd on th ndx that has th hghst R-squard wth th portfolo n quston. For st-ft MPT statstcs, Mornngstar compars th portfolo to dozns of dffrnt ndxs to fnd th st-ft. Both th road and st-ft rsults can usful to nvstors. Th road ndx R-squard can hlp nvstors dvrsfy thr portfolos. For xampl, an nvstor who alrady owns a fund wth a vry hgh corrlaton (and thus hgh R-squard) wth th S&P 500 mght not choos to uy anothr fund that corrlats closly to that ndx. Th st-ft MPT statstcs can hlp nvstors compar two smlar funds. For xampl, f two funds hav th sam st-ft ndx, an nvstor can valuat th rsk and xcss rturns for thos funds y comparng thr st-ft tas and alphas. Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 3

4 Mthodology Mornngstar calculats a fund s alpha, ta, and R-squard statstcs y runnng last-squars rgrsson of th fund s xcss rturn ovr Trasury lls compard wth th xcss rturns of th ndx that Mornngstar has slctd as th ndx for th fund s road asst class. Th Mornngstar road asst class ndxs ar as follows: Broad Asst Class Broad Asst Class Indx U.S. Stocks S&P 500 Intrnatonal Stocks MSCI EAFE Balancd Dow Jons Modrat Portfolo Taxal Bonds Lhman Brothrs Aggrgat Bond Muncpal Bonds Lhman Brothrs Mun Th calculatons ar mad usng th tralng 36-month prod. In last-squars rgrsson thr s dpndnt varal and an ndpndnt varal. A last squars rgrsson can undrstood y lookng at a scattr plot of th ndpndnt varal on th horzontal axs and th dpndnt varal on th vrtcal axs. Th rgrsson ln s th straght ln that mnmzs th sum of th squard vrtcal dstancs of ach scattr pont from th ln. To stmat th MPT statstcs for a fund usng ts road asst class ndx, Mornngstar runs a rgrsson wth th monthly xcss rturn on th fund as th dpndnt varal and th xcss rturn on th road asst class ndx as th ndpndnt varal. Th slop of th rsultng rgrsson ln s th stmat of ta and th ntrcpt (multpld y 1 to xprss t as an annual fgur) s th stmat of alpha. Y: Portfolo X: Bnchmark Indx Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 4

5 Bta Bta s a masur of a fund s snstvty to movmnts n th ndx. By constructon, th ta of th ndx s A fund wth a 1.10 ta has tndd to hav an xcss rturn that s 10% hghr than that of th ndx n up markts and 10% lowr n down markts, holdng all othr factors rman constant. A ta of 0.85 would ndcat that th fund has prformd 15% wors than th ndx n up markts and 15% ttr n down markts. A low ta dos not mply that th fund has a low lvl of volatlty, though; rathr, a low ta mans only that th fund s ndx-rlatd rsk s low. A spcalty fund that nvsts prmarly n gold, for xampl, wll usually hav a low ta (and a low R-squard), as ts prformanc s td mor closly to th prc of gold and gold-mnng stocks than to th ovrall stock markt. Thus, although th spcalty fund mght fluctuat wldly caus of rapd changs n gold prcs, ts ta wll low. In th xampls low, ta s 0.69 for th fund on th lft and 1.70 for th fund on th rght. Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 5

6 Bta (contnud) Bta s calculatd as: Cov β r = σ r whr: β = Bta of portfolo r r Cov = Covaranc twn th xcss rturns of th portfolo r and th nchmark r σ = Varanc of th xcss rturns of th nchmark and: Cov = 1 n r [( R )( R B B )] n -1 = 1 whr: R = Excss rturn of th portfolo for month = R - RF, whr R s th portfolo rturn for month and RF s th rsk-fr rturn for month R = Avrag monthly xcss rturn of th portfolo ovr n prods (smpl man) B = Excss rturn of th nchmark for month = B - RF, whr B s th nchmark rturn for month and RF s th rsk-fr rturn for month B = Avrag monthly xcss rturn of th nchmark ndx ovr n prods (smpl man) n = numr of prods (Mornngstar typcally uss 36 months) R s th smpl arthmtc avrag xcss rturn for th portfolo: R = 1 n n = 1 R Th dnomnator for ta s th varanc of th xcss rturns of th nchmark: σ = 1 n ( B B = 1 n -1 ) A smlar calculaton can also usd for th varanc of th portfolo, σ r. Standard dvaton s th squar root of varanc. Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 6

7 Alpha Alpha masurs a fund s prformanc aftr adjustng for th funds systmatc rsk as masurd y th fund s ta wth rspct to th ndx. An nvstor could hav formd a passv portfolo wth th sam ta that of th fund y nvstng n th ndx and thr orrowng or lndng at th rsk-fr rat of rturn. Alpha s th dffrnc twn th avrag xcss rturn on th fund and th avrag xcss rturn on th lvrd or d-lvrd ndx portfolo. For xampl, f th fund had an avrag xcss rturn of 6% pr yar and ts ta wth rspct to th S&P 500 was 0.8 ovr a prod whn th S&P 500 s avrag xcss rturn was 7%, ts alpha would 6% 0.8*7% = 0.4%. Thr ar lmtatons to alpha s alty to accuratly dpct a fund s addd or sutractd valu. In som cass, a ngatv alpha can rsult from th xpnss that ar prsnt n th fund fgurs ut ar not prsnt n th fgurs of th comparson ndx. Th usfulnss of alpha s compltly dpndnt on th accuracy of ta. If th nvstor accpts ta as a conclusv dfnton of rsk, a postv alpha would a conclusv ndcator of good fund prformanc. α M = R β B whr: α M = Monthly masur of alpha R = Avrag monthly xcss rturn of th portfolo B = Avrag monthly xcss rturn of th nchmark ndx Th rsultng alpha s n monthly trms, caus th avrag rturns for th portfolo and nchmark wr monthly avrags. Mornngstar thn annualzs alpha to put t n annual trms. 1 α A = 1 α M whr: α = Annualzd masur of alpha A 1 Pror to /8/005, Mornngstar annualzd th monthly alpha wth a gomtrc mthod, (1+α) 1-1. Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 7

8 R-squard R-squard s anothr statstc that s producd y a last-squars rgrsson analyss. R- squard s a numr twn 0 and 100% that masurs th strngth of th rlatonshp twn th dpndnt and ndpndnt varals. An R-squard of 0 mans that thr s no rlatonshp twn th two varals and an R-squard of 100% mans that th rlatonshp s prfct wth vry scattr pont fallng xactly on th rgrsson ln. Thus, stock ndx funds that track th S&P 500 ndx wll hav an R-squard vry clos to 100%. A low R-squard ndcats that th fund s movmnts ar not wll xpland y movmnts n th ndx. An R-squard masur of 35%, for xampl, mans that only 35% of th fund s movmnts can xpland y movmnts n th ndx. R-squard can usd to ascrtan th sgnfcanc of a partcular ta stmat. Gnrally, a hgh R-squard wll ndcat a mor rlal ta fgur. R-squard rangs from 0 (prfctly uncorrlatd) to 100 (prfctly corrlatd). Corrlaton ( ρ ) s th squar root of R-squard. R-squard s calculatd as follows: R = Cov r 100 ( ) σ σ r whr: Cov r = Covaranc twn th xcss rturns of th portfolo r and th nchmark ndx σ = Standard dvaton of th xcss rturns of th portfolo r r σ = Standard dvaton of th xcss rturns of th nchmark ndx Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 8

9 Bst-Ft Indx Mornngstar also shows addtonal alpha, ta, and R-squard statstcs asd on a rgrsson aganst th st-ft ndx. Th st-ft ndx for ach fund s slctd asd on th hghst R- squard rsult from sparat rgrssons on a numr of ndxs. For xampl, many hgh-yld funds show low R-squard rsults and thus a low dgr of corrlaton whn rgrssd aganst th road asst class ndx for th taxal ond funds, th Lhman Brothrs Aggrgat. Ths low R-squard rsults ndcat that th ndx dos not xplan wll th havor of most hghyld funds. Most hgh-yld funds, howvr, show sgnfcantly hghr R-squard rsults whn rgrssd aganst th CSFB Hgh-Yld Bond ndx. Both th road asst class and st-ft rsults can usful. Th road asst class ndx R- squard statstcs can hlp plan th dvrsfcaton of a portfolo of funds. For xampl, f an nvstor wshs to dvrsfy and alrady owns a fund wth a vry hgh corrlaton (and thus a hgh R-squard) wth th S&P 500, h or sh mght choos not to uy anothr fund that corrlats closly to that ndx. In addton, th st-ft ndx can usd to compar th tas and alphas of smlar funds that show th sam st-ft ndx. Mornngstar MPT Statstcs Novmr 30, Mornngstar, Inc. All rghts rsrvd. Th nformaton n ths documnt s th proprty of Mornngstar, Inc. Rproducton or transcrpton y any mans, n whol or part, wthout th pror wrttn consnt of Mornngstar, Inc., s prohtd. 9