Modern Portfolio Theory (MPT) Statistics

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1 Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or in part, without th prior writtn consnt of Morningstar, Inc., is prohibitd.

2 Introduction Modrn Portfolio Thory Statistics (MPT statistics) ar basd on th Capital Asst Pricing Modl (CAPM) of xpctd rturns dvlopd by Nobl laurat William Sharp and othrs in th arly 1960s. Th CAPM is basd on Modrn Portfolio Thory (MPT) dvlopd in th 1950s by Sharp s tachr and co-laurat Harry Markowitz. In th trminology of anothr Nobl laurat, th lat Milton Fridman, MPT is a normativ thory, maning that it is a prscription for how invstors ought to bhav. In contrast, th CAPM is a positiv thory in that it mant to b a dscription of how invstors do bhav. Th CAPM is basd on MPT in that it assums th all invstors follow th prscriptions of MPT. Th CAPM sparats th xcss rturn (i.., total rturn minus th rturn on a risk-fr scurity) of ach scurity into two componnts: systmatic xcss rturn and unsystmatic (or idiosyncratic) rturn. Systmatic xcss rturn is dirctly proportional to th xcss rturn of th markt portfolio. Th ratio of th xcss rturn of th markt to th systmatic xcss rturn of th scurity in qustion is th scurity s bta. Bta masurs how snsitiv th xcss rturn on a scurity is to th xcss rturn of th markt as a whol. On of th main implications of th CAPM is that th xpctd xcss rturn on a scurity is dirctly proportional to systmatic risk as masurd by bta and is not rlatd to any othr variabl. This mans that thr ar no rwards for taking on unsystmatic risk. In an fficint markt in which th CAPM holds, th only way to obtain an xpctd rturn abov that of th markt portfolio is to tak on a bta abov on. In th 1970 s, Michal Jnsn proposd a prformanc masur for activly managd funds that is basd on th CAPM calld Jnsn s alpha or simply alpha. Th ida of alpha is that a managr should not rciv crdit for achiving abov-markt prformanc by taking on systmatic risk as masurd by bta. Alpha is th avrag xcss rturn that a portfolio achivs abov and byond that could hav bn obtaind from position in th markt portfolio, lvrd or d-lvrd so as th hav th sam bta as th fund. 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd.

3 Introduction (continud) Strictly spaking, th CAPM cannot b applid in th ral world bcaus rturns on th markt portfolio ar unobsrvabl. To mak alpha and bta practical masurs at first, broad stock markt indxs wr usd as proxis for th markt portfolio. As funds bcam mor spcializd, mor narrow bnchmarks wr dvlopd to track rturns on th mor narrowly dfind sourcs of systmatic risk and rturn. Today it is common practic to masur alpha and bta using a narrowly dfind bnchmark that is chosn to rprsnt th main sourc of th systmatic risk of th fund bing analyzd. In addition to alpha and bta, a third MPT statistic is R-squard. R-squard masurs th strngth of th rlationship btwn xcss rturns on th bnchmark and xcss rturns on th fund bing analyzd. MPT statistics ar calculatd from a comparison of a fund s xcss rturns and th bnchmark s xcss rturns. Unlss a tim horizon is spcifid, Morningstar s MPT statistics ar basd on thr yars of monthly rturns. Morningstar calculats thr sts of MPT statistics for ach fund, using a standard st of bnchmarks for ach asst group; using a standard st of bnchmarks for ach Morningstar Catgory; and using th indx from th fund s prospctus. Morningstar also calculats bst-fit MPT statistics, which ar basd on th indx that has th highst R-squard with th portfolio in qustion. For bst-fit MPT statistics, Morningstar compars th portfolio to dozns of diffrnt indxs to find th bst-fit. Th broad asst class, catgory, prospctus and bst-fit rsults can all b usful to invstors. Th broad indx R-squard or catgory indx R-squard can hlp invstors divrsify thir portfolios. For xampl, an invstor who alrady owns a fund with a vry high corrlation (and thus high R-squard) with th S&P 500 might not choos to buy anothr fund that corrlats closly to that indx. Th bst-fit MPT statistics can hlp invstors compar two similar funds. For xampl, if two funds hav th sam bst-fit indx, an invstor can valuat th risk and xcss rturns for thos funds by comparing thir bst-fit btas and alphas. 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 3

4 Mthodology Morningstar calculats a fund s alpha, bta, and R-squard statistics by running last-squars rgrssion of th fund s xcss rturn ovr a risk-fr rat compard with th xcss rturns of th indx that Morningstar has slctd as th indx for th fund s broad asst class or th fund s catgory indx. Th Morningstar broad asst class indxs for th US ar as follows: Broad Asst Class Broad Asst Class Indx U.S. Stocks S&P 500 Intrnational Stocks MSCI EAFE Balancd Morningstar Modrat Targt Risk Altrnativ ML USD LIBOR 3 Mon Taxabl Bonds BarCap US Aggrgat Bond Municipal Bonds BarCap Municipal Morningstar s ditorial tam assigns th catgory indx. Th chosn indx rprsnts th bst fit (has th highst corrlation to funds in th catgory) of thos indxs that hav at last thr yars worth of rturn history availabl (for nwr catgoris) and at last 10 yars worth of rturn history availabl (for xisting catgoris). Th catgory indx assignmnts for Europ/Asia can b found in th mthodology documnt Morningstar Catgory Dfinitions Europ and Asia. Th catgory indx assignmnts for th US can b found in th mthodology documnt Morningstar Catgory Classifications. Th prospctus indx is th indx (a.k.a. bnchmark) that th fund dfins in its prospctus. A high corrlation to this indx suggsts that th fund has chosn an appropriat indx against which to valuat its prformanc. A low corrlation may indicat that th fund chos an indx that may not b vry rprsntativ of th fund s invstmnt styl. Th calculations ar mad using th trailing 36-month priod. In last-squars rgrssion thr is dpndnt variabl and an indpndnt variabl. A last squars rgrssion can b undrstood by looking at a scattr plot of th indpndnt variabl on th horizontal axis and th dpndnt variabl on th vrtical axis. Th rgrssion lin is th straight lin that minimizs th sum of th squard vrtical distancs of ach scattr point from th lin. To stimat th MPT statistics for a fund using its broad asst class indx, Morningstar runs a rgrssion with th monthly xcss rturn on th fund as th dpndnt variabl and th xcss rturn on th broad asst class indx as th indpndnt variabl. Th slop of th 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 4

5 Mthodology (continud) rsulting rgrssion lin is th stimat of bta and th intrcpt (multiplid by 1 to xprss it as an annual figur) is th stimat of alpha. Y: Portfolio X: Bnchmark Indx 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 5

6 Bta Bta is a masur of a fund s snsitivity to movmnts in th indx. By construction, th bta of th indx is A fund with a 1.10 bta has tndd to hav an xcss rturn that is 10% highr than that of th indx in up markts and 10% lowr in down markts, holding all othr factors rmain constant. A bta of 0.85 would indicat that th fund has prformd 15% wors than th indx in up markts and 15% bttr in down markts. A low bta dos not imply that th fund has a low lvl of volatility, though; rathr, a low bta mans only that th fund s indx-rlatd risk is low. A spcialty fund that invsts primarily in gold, for xampl, will usually hav a low bta (and a low R-squard), as its prformanc is tid mor closly to th pric of gold and gold-mining stocks than to th ovrall stock markt. Thus, although th spcialty fund might fluctuat wildly bcaus of rapid changs in gold prics, its bta will b low. In th xampls blow, bta is 0.69 for th fund on th lft and 1.70 for th fund on th right. 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 6

7 Bta (continud) Bta is calculatd as: Cov β r = σ rb b whr: β r = Bta of portfolio r Cov rb = Covarianc btwn th xcss rturns of th portfolio r and th bnchmark b σ = Varianc of th xcss rturns of th bnchmark b and: Cov = 1 n rb [( R )( i R Bi B )] n -1 i= 1 whr: R i = Excss rturn of th portfolio for month i = R i - RF i, whr R i is th portfolio rturn for month i and RF i is th risk-fr rturn for month i R = Avrag monthly xcss rturn of th portfolio ovr n priods (simpl man) B i = Excss rturn of th bnchmark for month i = B i - RF i, whr B i is th bnchmark rturn for month i and RF i is th risk-fr rturn for month i B = Avrag monthly xcss rturn of th bnchmark indx ovr n priods (simpl man) n = numbr of priods (Morningstar typically uss 36 months) R is th simpl arithmtic avrag xcss rturn for th portfolio: R 1 = n n i= 1 R i Th dnominator for bta is th varianc of th xcss rturns of th bnchmark: σ b = 1 n ( B i B i= 1 n -1 ) A similar calculation can also b usd for th varianc of th portfolio, σ r. Standard dviation is th squar root of varianc. 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 7

8 Alpha Alpha masurs a fund s prformanc aftr adjusting for th funds systmatic risk as masurd by th fund s bta with rspct to th indx. An invstor could hav formd a passiv portfolio with th sam bta that of th fund by invsting in th indx and ithr borrowing or lnding at th risk-fr rat of rturn. Alpha is th diffrnc btwn th avrag xcss rturn on th fund and th avrag xcss rturn on th lvrd or d-lvrd indx portfolio. For xampl, if th fund had an avrag xcss rturn of 6% pr yar and its bta with rspct to th S&P 500 was 0.8 ovr a priod whn th S&P 500 s avrag xcss rturn was 7%, its alpha would b 6% 0.8*7% = 0.4%. Thr ar limitations to alpha s ability to accuratly dpict a fund s addd or subtractd valu. In som cass, a ngativ alpha can rsult from th xpnss that ar prsnt in th fund figurs but ar not prsnt in th figurs of th comparison indx. Th usfulnss of alpha is compltly dpndnt on th accuracy of bta. If th invstor accpts bta as a conclusiv dfinition of risk, a positiv alpha would b a conclusiv indicator of good fund prformanc. α M = R β B whr: α M = Monthly masur of alpha R = Avrag monthly xcss rturn of th portfolio B = Avrag monthly xcss rturn of th bnchmark indx Th rsulting alpha is in monthly trms, bcaus th avrag rturns for th portfolio and bnchmark wr monthly avrags. Morningstar thn annualizs alpha to put it in annual trms. 1 whr: α = Annualizd masur of alpha A α A = 1 α M 1 Prior to /8/005, Morningstar annualizd th monthly alpha with a gomtric mthod, (1+α) Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 8

9 R-squard R-squard is anothr statistic that is producd by a last-squars rgrssion analysis. R- squard is a numbr btwn 0 and 100% that masurs th strngth of th rlationship btwn th dpndnt and indpndnt variabls. An R-squard of 0 mans that thr is no rlationship btwn th two variabls and an R-squard of 100% mans that th rlationship is prfct with vry scattr point falling xactly on th rgrssion lin. Thus, stock indx funds that track th S&P 500 indx will hav an R-squard vry clos to 100%. A low R-squard indicats that th fund s movmnts ar not wll xplaind by movmnts in th indx. An R-squard masur of 35%, for xampl, mans that only 35% of th fund s movmnts can b xplaind by movmnts in th indx. R-squard can b usd to ascrtain th significanc of a particular bta stimat. Gnrally, a high R-squard will indicat a mor rliabl bta figur. R-squard rangs from 0 (prfctly uncorrlatd) to 100 (prfctly corrlatd). Corrlation ( ρ ) is th squar root of R-squard. R-squard is calculatd as follows: R = Cov rb 100 ( ) σ σ r b whr: Cov = r rb Covarianc btwn th xcss rturns of th portfolio r and th bnchmark indx b σ = Standard dviation of th xcss rturns of th portfolio r σ = Standard dviation of th xcss rturns of th bnchmark indx b b 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 9

10 Bst-Fit Indx Morningstar also shows additional alpha, bta, and R-squard statistics basd on a rgrssion against th bst-fit indx. Th bst-fit indx for ach fund is slctd basd on th highst R- squard rsult from sparat rgrssions on a numbr of indxs. For xampl, many high-yild funds show low R-squard rsults and thus a low dgr of corrlation whn rgrssd against th broad asst class indx for th taxabl bond funds, th Lhman Brothrs Aggrgat. Ths low R-squard rsults indicat that th indx dos not xplain wll th bhavior of most highyild funds. Most high-yild funds, howvr, show significantly highr R-squard rsults whn rgrssd against th CSFB High-Yild Bond indx. Th broad asst class, catgory, prospctus, and bst-fit rsults can all b usful to invstors. Th broad asst class indx or th catgory indx R-squard statistics can hlp plan th divrsification of a portfolio of funds. For xampl, if an invstor wishs to divrsify and alrady owns a fund with a vry high corrlation (and thus a high R-squard) with th S&P 500 or th FTSE 100, h or sh might choos not to buy anothr fund that corrlats closly to that indx. In addition, th bst-fit indx can b usd to compar th btas and alphas of similar funds that show th sam bst-fit indx. Th prospctus indx R-squard statistics can hlp dtrmin whthr th portfolio managr has chosn an appropriat indx to b masurd against. 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription by any mans, in whol or part, without th prior writtn consnt of Morningstar, Inc., is prohibitd. 10

Modern Portfolio Theory (MPT) Statistics

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