Volum 61 Numbr 5 005, CFA Institut PERSPECTIVES Why Markt-Valuation-Indirnt Indxing Works Jack Trynor By th nd o th 0th cntury, vn casual invstors had bcom comortabl with th ida o indx unds. Th ida o a bttr indx und (s Arnott, Hsu, and Moor 005), howvr, is mind-boggling. This articl ors on man s viw o why it will actually work. H dins markt-valuation-indirnt (MVI) indxing to b indxing in which th indx is built on any wights that avoid th roblm with markt caitalization. Th bad nws about stock markts is that thy ric stocks imrctly. Th good nws is that th misricings ar always rlativ. Not only will ovrricd stocks b countrbalancd by undrricd stocks, but th distribution o rror at any oint in tim will b symmtrical. W can ictur this distribution as a bll-shad curv with rror on th horizontal axis and som masur o rquncy on th vrtical axis. Bcaus it rlcts both th numbr o comanis and thir siz, aggrgat valu is th aroriat masur o rquncy. But which masur o aggrgat valu tru valu or markt valu? I w us markt valu, thn, alas, it will mak biggr bts on ovrricd stocks and smallr bts on undrricd stocks. To gt a handl on how much rror, w bgin by dining u rlativ rror (xrssd as a raction o tru valu) and v(u) amount o tru valu with rror. Whn w considr th thousands o stocks in th markt, th randomnss o articular stocks is submrgd in a dnsity unction that associats a rlativly stabl amount o dnsity unction v(u) with rlativ rror u to satisy v( u) v( u). (1) But 1 + u is th markt valu o $1.00 o tru valu with rlativ rror u. So th amount o markt Jack Trynor is rsidnt and CEO o Trynor Caital Managmnt, Inc., Palos Vrds Estats, Caliornia. valu with rror u is (1 + u)v(u); thn, th rror distribution satisis ( 1+ u) v( u) > ( 1 u) v( u). Unlik th distribution o th ricing rror that uss tru valu, th rror distribution or markt valus is skwd to th right. This lack o symmtry is th roblm with caitalization wighting: By using markt valus to dtrmin its wights, a ca-wightd indx und will invst mor mony in ovrricd stocks than in undrricd stocks. Considr a symmtrical distribution o markt rrors u around a man rror u. For ach stock whos rror xcds th man by u u, thr will tnd to b a stock whos rror alls short o th man by u u. Exrssd in trms o a rquncy unction v( ) o tru valus, th original symmtry condition is obviously satisid by v( u u) v( u u), () (3) bcaus th scond argumnt is indd minus th irst, as w sciid. On th othr hand, i w xct markt rrors to b symmtrical around a man rror o zro, w nd to add th ollowing condition: wightd by th tru valus, th man o th rrors in markt ric is zro. In trms o our symbols, w can xrss th nw condition: uv ( u) 0. (4) Obviously, th sum ovr all stocks undrricd and ovrricd is zro. Th Basic Equation How dos MVI indxing avoid ca-wightd indxing s roblm? Th ky is a siml quation linking th covariancs o ortolio wights with markt ric r shar, tru valu r shar, and rrors in markt ric r shar. Stmbr/Octobr 005 www.caubs.org 65 Coyright 005, CFA Institut. Rroducd and rublishd rom Financial Analysts Journal with rmission rom CFA Institut. All rights rsrvd.
I, as bor, u is th rlativ rror, thn 1 + u is th ratio o markt to tru valu v and v( 1 + u) v+ vu (5) is markt ric. So, to a common divisor qual to th numbr o stocks, th covarianc o ortolio wights w with shar rics is wv( 1+ u) w u( 1+ u) wv + wvu w v wv w v + wvu. (6) Th xrssion in brackts is th covarianc o ortolio wights with th tru shar valus. Now, considr th covarianc o ortolio wights with dollar rrors in shar ric, cov w( uv) wuv w uv wuv cov w( uv) + w uv. (7) W s that th xrssion wuv quals this covarianc lus th roduct wuv. But undr our xandd symmtry condition, w hav uv w uv 0. (8) So th irst o th thr covariancs quals th algbraic sum o th scond and third. Th scond is th covarianc o ortolio wights with tru valus, and th third is th covarianc o th wights with th dollar rrors in rics. Imlications or MVI Indxing On alication o MVI indxing is wighting schms in which th covarianc o wights with markt valus is zro. In this cas, to satisy Equation 6, ithr th othr two covariancs must ost xactly which is highly imrobabl or both must b zro. An xtrm xaml is a ortolio with qual wights. On avrag, th numbr o ovrricd stocks will b th sam as th numbr o undrricd stocks. But i all th stocks ar assignd th sam wight, th invstmnt in th ovrricd sgmnt will dnd only on that numbr and th invstmnt in th undrricd sgmnt will dnd only on that sam numbr. So th two invstmnts will tnd to b qual in contrast to th ca-wightd indx und, which ays mor or th ovrricd sgmnt and lss or th undrricd sgmnt. Alas, a schm that wights largca and small-ca stocks th sam is going to hav small-ca markt bias, howvr, rlativ to many bnchmark ortolios, hnc mor snsitivity to any systmatic small-ca actor (as discussd in, or xaml, Fama and Frnch 1973). Th quation rlating th thr covariancs can b alid in othr ways. For xaml, instad o dmonstrating mirically that a givn st o wights has zro covarianc with markt rics, w can aal to a riori rasons why crtain sts o wights will hav a zro covarianc with th rrors. W hav sn that i th ortolio givs th sam wight to undrricing rrors it givs to ovrricing rrors, th third covarianc vanishs. But thn th othr two covariancs in th quation must b qual. So w can us markt valus, which ar obsrvabl, rathr than tru valus, which ar not, to stimat th small-ca bias in such sts o wights. Eliminating Small-Ca Bias Is th constant-wight ortolio th bst MVI indxing can do? Dos it hav th smallst tracking rror vrsus a convntional ca-wightd indx? Som wighting schms will hav lss small-ca bias than othrs. Examls includ wighting by numbr o mloys, numbr o customrs, or sals. And som schms may actually wight larg cas mor havily than th markt indxs do. Suos w usd th numbr o cororat jts or cororat limousins. Radrs ar ncouragd to giv r rin to thir imagination. A dirnt aroach is to rank stocks by caitalization. Form ca-wightd ortolios that start with th biggst singl stock, th biggst stocks, tc., u to 500 stocks. Evry on o ths ortolios xct th last will hav a larg-ca bias rlativ to th S&P 500 Indx. But ach MVI ortolio will hav a small-ca bias rlativ to its corrsonding cawightd countrart. Thus thr will always b a uniqu numbr o stocks or which th MVI ortolio has th sam small-ca bias as th cawightd S&P 500. I this brakvn ortolio includs nough stocks, it can still b satisactorily divrsiid. W hav still othr ways to rmov small-ca bias. Considr a ca-wightd ortolio o th 100 smallst comanis in th Wilshir 5000 Indx. It will hav no alha rsulting rom MVI indxing and lots o small-ca bias. A short osition in this ortolio will ost a lot o small-ca bias without rducing th MVI alha. 66 www.caubs.org 005, CFA Institut
Why Markt-Valuation-Indirnt Indxing Works An aroriat blnd o any two schms with oosit biass will always liminat bias rlativ to any givn bnchmark. And i dirnt clints hav dirnt bnchmarks, th blnd can b tailord to thir bnchmarks. Th Sourc o MVI s Advantag Stocks in th MVI ortolio with a givn tru valu may gt a larg wight or a small wight. Bcaus thy ar as likly to b undrricd as ovrricd, howvr, whatvr wight th mthod assigns is as likly to contribut to th undrricd stock as to th ovrricd stock. Avragd across all th stocks in th MVI ortolio, th aggrgat dollar invstmnts will tnd to b th sam. O cours, at a oint in tim, ral stocks won t oblig th author by alling into xactly countrbalancing airs. But th asist way to xlain how MVI caitalizs on th tndncy or ricing rrors to b symmtric is to ocus on such an idalizd air. Bcaus o th rrors in markt ric, th corrsonding undrricd or ovrricd stocks in a ca-wightd ortolio will hav dirnt markt valus vn i thy hav th sam tru valus. Lt th tru valus o thos stocks b v, and lt th aggrgat ricing rrors b + and. I ca-wightd invstors snd v + dollars on th ormr and v dollars on th lattr, thy will snd a total o ( v + ) + ( v ) v dollars and gt (9) v v ( v + ) ( v ) v (10) + + worth o tru valu. On th othr hand, th MVI invstors snd th sam numbr o dollars on th undrricd as thy snd on th ovrricd stocks. But a dollar snt on ovrricd scuritis buys lss tru valu than a dollar snt on undrricd scuritis. For xaml, a dollar snt on a stock with tru valu v and markt ric v + buys v/(v + ) o th tru valu; a dollar snt on a stock with tru valu v and markt ric v buys v/(v ) o th tru valu. I th MVI invstors snd v dollars on ach stock, thy mak th sam total invstmnt as th ca-wightd invstors and gt v v v v + + worth o tru valu, or v v ( v ) + v ( v + ) v v v v v v v (11) 1 v, v 1 whr th xrssion in brackts is always gratr than zro. (Th xrssion /ν is what w rviously calld u ric rror rlativ to tru valu.) Thus or th sam total invstmnt, th MVI invstors own mor tru valu than th cawightd invstors, with a dirnc that dnds only on th rlativ siz o th aggrgat ricing rror. Th gain or th whol markt sums across rrors occurring with a wid rang o rquncis. I th rquncy unction is (/ν), thn th gain can b xrssd as ( / v) d( / v). (1) 1 ( / v) For small rrors, w can aroximat this intgral by 1 + v v d v v d v + v v d v. (13) Th valu o th irst intgral is 1. I, as w hav assumd or th rquncy distribution o tru valus, th man o th rrors is 0, thn th scond intgral is th varianc o th rrors. Whn stocks ar accuratly ricd, th MVI ortolio ralizs no gain rlativ to th ricwightd ortolio. But whn th rror in markt rics is xrssd as a raction o th tru valu, thn th gain rom MVI is th squar o th standard rror, σ. Tabl 1 dislays a rang o ossibl valus o σ, σ, 1 + σ, and (or rasons to b xlaind) 1/(1 σ ). MVI invstors raliz this bnit vn i misricd stocks nvr rvrt to thir tru valus. I rvrsion occurs, it ors an additional bnit (s Andix A). To b sur, th corrct intgral is not as simly rlatd to th standard rror o stock rics as our crud aroximation is. But in th vnt, small ricing rrors will b much mor rqunt than larg ricing rrors. Th radr can gt som sns o how bad our aroximation is by imagining that, instad o bing saml avrags, th numbrs in Stmbr/Octobr 005 www.caubs.org 67
Tabl 1. MVI s Advantag or Indicatd Standard Errors in Markt Pric 1 ----------- σ σ 1 + σ 1 σ 0.01 0.0001 1.0001 1.0001 0.0 0.0004 1.0004 1.0004 0.04 0.0016 1.0016 1.0016 0.08 0.0064 1.0064 1.0064 0.1 0.0144 1.0144 1.0146 0.14 0.0196 1.0196 1.000 0.16 0.056 1.056 1.063 0.18 0.034 1.034 1.0335 0.0 0.0400 1.0400 1.0417 0. 0.0484 1.0484 1.0509 0.4 0.0576 1.0576 1.0611 0.6 0.0676 1.0676 1.075 0.8 0.0784 1.0784 1.0851 0.30 0.0900 1.0900 1.0989 0.3 0.104 1.104 1.1141 0.34 0.1156 1.1156 1.1307 0.36 0.196 1.196 1.1489 0.38 0.1444 1.1444 1.1688 0.40 0.1600 1.1600 1.1905 0.4 0.1764 1.1764 1.14 0.44 0.1936 1.1936 1.401 0.46 0.116 1.116 1.684 0.48 0.304 1.304 1.994 0.50 0.500 1.500 1.3333 th σ column ar ric rrors on a sciic stock, in which that stock s contribution to th aroximation rror is th dirnc btwn th righthand columns. It taks a 31 rcnt ricing rror to roduc a 1 rcnt rror in such a stock s contribution to th intgral. And all individual stock rrors, small or larg, ositiv or ngativ, caus th author s aroximation to undrstat th tru gain rom MVI. But that s th only uros in including th right-hand column. Th author trusts nobody will think it is an stimat o th tru valu o th intgral or th indicatd varianc. Bcaus w can t obsrv th markt s ricing rrors, w can t radily rsolv dbats about thir magnitud. Eugn Fama has on viw; Fischr Black had anothr. A 1 rcnt standard rror in stock rics roducs a gain rlativ to ca wighting o 0.0001 surly too small to warrant intrst in MVI wighting. But th gain incrass raidly as th standard rror incrass, bing 400 tims as big or a 0 rcnt standard rror. Can w aord to b wrong about our rconctions? Trading Costs MVI ortolio managrs trad mor than managrs o ca-wightd ortolios, although how much mor dnds on th ric discrancis th MVI managrs choos to tolrat bor trading back to th rscribd wights. Th trad siz will incras with t, so volum will b roortional to siz t t 1 1. t t siz (14) Th cost o incrasing th triggr siz is dartur rom th ortolio roortions rscribd by MVI. Trading lags bring MVI closr to th ca-wightd rsult. Whn all rics ris or all in roortion to th MVI ortolio managr s wights, howvr, no trading is ndd. Conclusion Th author has argud that on dosn t nd to know tru valus in ordr to avoid th roblm with ca-wightd indx unds. On can still njoy all th bnits o an indx und a high lvl o divrsiication and low trading costs by invsting randomly with rsct to th markt s ricing rrors. Andix A: Rvrsion to Tru Valu Th rat o rturn rom th rvrsion o markt valu to tru valu dnds on th rvrsion rat. Is th avrag tim to rvrsion 1 yar or 10 yars? W do not know. Prsumably, rsulting rats o rturn ar also roortional to th initial ricing rror. Assum ovr- and undrricd stocks hav th sam absolut rror ; thn, or an ovrricd stock with tru valu v 1 and markt ric 1, th rat o rturn is roortional to v1 1 (A1), 1 1 and or an undrricd stock with tru valu v and markt ric, th rat o rturn is roortional to v. (A) 68 www.caubs.org 005, CFA Institut
Why Markt-Valuation-Indirnt Indxing Works For th MVI invstor with qual ositions in th two stocks, th avrag rturn is 1 1 1 1 1 1 1 1 v 1 (A3) bor dividing by th ctiv rvrsion tim. For th whol ortolio, th rturn is 1 1 (A4) T v v d v var T v, again assuming a man o zro. Rrncs Arnott, Robrt D., Jason Hsu, and Phili Moor. 005. Fundamntal Indxation. Financial Analysts Journal, vol. 61, no. (March/Aril):83 99. Fama, Eugn, and Knnth Frnch. 1973. Th Cross-Sction o Exctd Rturns. Journal o Financ, vol. 47, no. (Jun):47 465. Stmbr/Octobr 005 www.caubs.org 69