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3 THE VALUE SPREAD The rao of book value o marke value of common equy (.e., he book-o-marke rao) s an mporan deermnan of he cross-secon of average equy reurns. Frms wh hgh book-o-marke raos have hgher average reurns han frms wh low book-o-marke raos [Rosenberg, Red, and Lansen (1985), Fama and French (1992), and ohers]. Thus he book-o-marke raos have nformaon concernng subsequen expeced reurns. Smulaneously, dfferences n frms book-o-marke raos are also relaed o dfferences n fuure expeced cash-flow and earnngs growh as well as fuure profably. Low-book-omarke frms grow faser and are perssenly more profable han hgh-book-o-marke frms. Inuvely, boh expeced reurns and expeced growh play a role n deermnng he marke prce and hus he book-omarke rao. We decompose he cross-seconal varance of frms book-o-marke raos usng a long ( ) U.S. panel and a shorer ( ) nernaonal panel. Our varance decomposon shows wha fracon of he cross-seconal dsperson n he book-o-marke raos s caused by varaon n expeced reurns and wha fracon by varaon n expeced cash flows. Provded ha book-o-marke does no behave explosvely, an approxmae deny equaes he curren book-o-marke rao wh an nfne dscouned sum of fuure sock reurns and profably. Ths deny mples ha a low curren book-o-marke rao has o be usfed by eher hgh expeced fuure profably or low expeced fuure reurns. Therefore, o a very accurae approxmaon, all cross-seconal varaon n he book-o-marke raos mus necessarly be accouned for by cross-seconal varaon n expeced long-horzon sock reurns and/or profably. The observaon ha he cross-seconal varaon n he book-o-marke raos s relaed o crossseconal varaon n fuure profably s no new. For example, Fama and French (1995) show ha a hgh book-o-marke rao sgnals perssen poor earnngs and profably and a low book-o-marke rao sgnals srong earnngs and profably. The novel par of our analyss s ha by usng a presen-value model we are able o measure exacly how much he cross-seconal varaon n expeced profably conrbues o he cross-seconal varaon n frms' book-o-marke raos. Therefore, our analyss s able o quanavely, no us qualavely, commen on he source of he heerogeney n valuaon raos across frms.

4 Our frs and mos basc fndng s easly summarzed. We fnd ha ransory varaon n 15-year expeced reurns s responsble for only a small fracon of he cross-seconal book-o-marke varance. In he long U.S. panel, approxmaely 8% of he uncondonal cross-seconal varance of he book-o-marke raos can be explaned by expeced fuure 15-year profably and perssence of valuaon levels, whle us 2% can be explaned by ransory varaon n expeced reurns. Ths fracon appears sable across me and across ypes of socks. I s neresng o relae our basc cross-seconal resul o aggregae me-seres resuls n he prevous leraure. Earler sudes [Campbell and Shller (1988), Cochrane (1992), Vuoleenaho (2), and ohers] examne he me seres of aggregae scaled-prce measures o quanfy he naure of he nformaon hey conan. These papers fnd ha he subsanal maory of he nformaon n he me seres of scaled prces s abou expeced reurns, and relavely lle s abou cash flows. For example, Vuoleenaho (2) fnds ha almos all of he me-seres varaon n he aggregae book-o-marke rao s due o expeced sock reurns and approxmaely none due o expeced profably. Our observaon ha he cross-secon of ndvdual socks looks very dfferen from aggregae ndces n hs regard s neresng n s own rgh bu may also be relevan for he nerpreaon of he prevous aggregae sudes. If he crosssecon of valuaon raos s largely drven by raonal cash-flow expecaons, he concluson ha he aggregae valuaon raos are exclusvely drven by rraonal senmen s perhaps less plausble. Our resuls also sugges ha mos of a growh sock s aypcal valuaon s smply due o hgh expeced profably raher han due o ha sock havng a low expeced reurn. For example, Csco Sysems and General Moors have approxmaely he same book value (Csco $29.7 vs. GM $31.5 bllon) bu Csco has a far greaer marke value (Csco $175.6 vs. GM $31.3 bllon). Accordng o our resuls, mos of he dfference s because equy on Csco s books s expeced o be more producve han ha on GM s. 1 However, some reverson oward book value n he form of hgher relave sock reurns for GM s o be expeced. Noe ha our basc decomposon s slen as o wheher he nformaon we measure n a frm's book-o-marke rao concernng subsequen reurns s due o rsk or msprcng. In he above example, for example, he hgher expeced equy reurns for GM relave o Csco may or may no be due o msprcng of eher sock; he dfference may be rsk-relaed. However, even f one akes he exreme sand and characerzes he expeced-reurn componen of book-o-marke as a marke-neffcency phenomenon, our 2

5 evdence suggess ha a mos abou 2% of he varaon n valuaon raos across frms can be lnked o capal-marke neffcences. Mos of ha varaon represens dfferences n expeced fuure profably, heerogeney ha s consderably less conroversal. We refne our basc resul by exendng our analyss of he cross-seconal heerogeney of valuaon raos. Frs, we replcae our measuremens usng a shorer nernaonal panel ( , 23 counres excludng he Uned Saes). Whle he shor me dmenson of our nernaonal sample reduces he sascal precson of our esmaes, he nernaonal pon esmaes suppor he conclusons drawn from he U.S. sample. A he fve-year horzon, approxmaely 41% of he varaon n he counry-adused booko-marke raos can be allocaed o expeced counry-adused profably, 14% o expeced counryadused sock reurns, and he remanng 47% o perssence of he counry-adused book-o-marke raos. The fac ha we oban smlar pon esmaes from a largely ndependen sample lends credbly o our basc resul. Second, n order o connue o sudy he relaon beween he resuls a dfferen levels of aggregaon, we furher spl he nformaon n book-o-marke concernng fuure reurns and fuure profably no nra- and ner-ndusry componens. We fnd ha he book-o-marke effec n reurns s mosly an nra-ndusry effec. Ths confrms evdence a he monhly horzon presened by Cohen and Polk (1999), Lewellen (1999), Asness, Porer, and Sevens (2). More mporanly, hese resuls show ha conrollng for ndusry effecs does no aler our varance decomposon resuls. Thrd, we documen sascally srong resuls concernng he predcably of reurns on sraeges ha go long value socks and shor growh socks. The above varance-decomposon resuls sugges ha he dfference n average reurns beween value and growh socks may vary hrough me wh he value spread (.e., he book-o-marke rao of a ypcal value sock mnus ha of a ypcal growh sock). In oher words, f he varance decomposon s consan over me wh a non-rval poron of he varance allocaed o expeced reurns, he level of he value spread should predc reurns on value mnus growh sraeges. Socks whose book equy s cheap should have especally hgh expeced reurns a mes when her book equy s especally cheap. We es hs hypohess on he Fama-French (1993) HML porfolo and fnd ha he HML value spread ndeed has srong predcve power for HML reurns, evdence ha he expeced reurn on he HML porfolo vares over me. Our expeced-reurn esmaes sugges ha he expeced reurn on HML was zero 3

6 or negave n he perods and Pon esmaes obaned from an nernaonal analyss are conssen wh he U.S. HML-predcably resuls. Our fndngs are also conssen wh hose of Asness, Fredman, Kral, and Lew (2) who examne he cross-secon of scaled-prce measures and reurns on scaled-prce porfolos. Usng daa from a relavely recen perod ( ) and a sample of U.S. socks lsed on I/B/E/S, hey fnd ha value spreads and spreads n proeced earnngs growh have predcve power for he me-seres of monhly reurns on value versus growh sraeges. The remander of he paper s organzed as follows. Secon I descrbes he daa. Secon II presens he varance-decomposon framework and resuls. Secon III concenraes on predcng value-mnusgrowh reurns. Secon IV concludes. I. Daa A. U.S. panel daa se The basc U.S. daa come from he merger of hree daabases. The frs one of hese, he Cener for Research n Secures Prces (CRSP) monhly sock fle, conans monhly prces, shares ousandng, dvdends and reurns for NYSE, AMEX, and NASDAQ socks. The second daabase, he COMPUSTAT annual research fle, conans he relevan accounng nformaon for mos publcly raded US socks. The COMPUSTAT accounng nformaon s supplemened by he hrd daabase, Moody s book equy nformaon colleced by Davs, Fama, and French (2). 2 The basc merged daa covers he perod , bu we only nclude he perod n our long U.S. panel. (We also oban our man saed resuls n he daa se ha ncludes he early years.) In he fnal sample, all varables are of annual frequency and he panel conans 192,661 frm-years. Appendx 1 conans dealed daa defnons. Alhough he basc daa se has some daa as early as year 1928, we begn our long U.S. panel daa se a he end of year 1937 n order o ensure a meanngful quanave nerpreaon of he book-o-marke and profably seres. The logc behnd hs choce s based on dsclosure regulaon. Before he Secures Exchange Ac of 1934, here was essenally no regulaon o ensure he flow of accurae and sysemac accounng nformaon. Among oher hngs, he ac prescrbes specfc annual and perodc reporng and record-keepng requremens for hese companes. The companes requred o fle repors wh he SEC mus also "make and keep books, records, and accouns, whch, n reasonable deal, accuraely and farly reflec he ransacons and dsposon of he asses of he ssuer. In addon, he legslaon nroduces he 4

7 concep of an ndependen publc or cerfed accounan o cerfy fnancal saemens and mposes sauory lables on accounans. Whle he value-relevan nformaon may have been dsclosed va oher channels, s clear ha nerpreng he pre-1934 and pos-1934 book-o-marke raos as havng smlar nformaon conen would be unrealsc. Afer sudyng he mplemenaon of he ac, we also decded o exclude he daa segmen. Ths perod could reasonably be characerzed as an nal enforcemen perod, afer whch reporng convenons have converged o her seady saes. 3 B. Inernaonal panel daa se In addon o he long U.S. panel, we also consruc a shorer nernaonal panel. The annual nernaonal panel s consruced from a monhly daa se creaed by Granham, Mayo, Van Oerloo & Co. (GMO). GMO s monhly daa se s n urn consruced from he Morgan Sanley Capal Inernaonal daa and varous real-me daa sources. The nernaonal panel spans he perod The number of counres n he daa se begns a 19 and ends a 22, and 23 dfferen counres are ncluded n he panel a varous pons of me. U.S. daa are no ncluded. The nernaonal panel conans 27,913 frm-years. II. Decomposng he cross-seconal varance of frms book-o-marke raos I s well known ha frms book-o-marke (BE/ME) raos descrbe cross-seconal varaon n reurns and profably. For example, Fama and French (1995) show ha value socks ypcally have hgher sock reurns and lower profably han growh socks. However, calculang how each of hese wo effecs conrbue o he cross-seconal spread n frms' BE/ME raos requres a quanave framework. We employ Vuoleenaho s (2) reurn-profably model o decompose he cross-seconal varance of BE/ME raos no hree componens: 1) covarance of fuure marke-adused sock reurns wh pas marke-adused BE/ME raos, 2) covarance of fuure marke-adused profably (.e., accounng reurn on equy ROE) wh pas marke-adused BE/ME raos, and 3) perssence of marke-adused BE/ME raos. The followng nuon underles he decomposon. Suppose wo frms (neher one of whch pays dvdends or ssues equy) have dfferen BE/ME raos. Over me, hs value spread can close by eher hgh BE/ME frms experencng hgher sock reurns han low BE/ME frms or by hgh BE/ME frms beng less profable and growng her book equy slower han he low BE/ME frms. On he one hand, f he 5

8 spread closes va sock reurns, he covarance of pas relave BE/ME raos and fuure relave reurns s posve. On he oher hand, f he spread closes va profably, he covarance of pas relave BE/ME raos and fuure relave profably s negave. There s also a hrd possbly he spread may no close compleely. In hs case, he dfference n BE/ME raos persss, and he covarance of pas and fuure relave BE/ME raos s hgh. A. Approxmae model of he book-o-marke rao To derve Vuoleenaho s (2) reurn-profably model necessaes hree specfc assumpons. Frs, because he model s saed n erms of logarhms, one mus assume book value, BE, marke value, ME, and dvdends, D, are srcly posve. Alhough hs assumpon s no necessarly sasfed for all ndvdual frms, s almos surely sasfed for he BE/ME-sored porfolos we use n our ess. Second, one mus assume frms' log BE/ME raos o be saonary, even hough boh log book and marke equy seres have an negraed componen. Thrd, we assume ha earnngs, dvdends, and book equy seres sasfy he clean-surplus relaon. 4 The clean-surplus relaon es he ncome saemen and balance shee dynamcs ogeher. In ha relaon, earnngs, dvdends, and book equy sasfy: BE BE 1 = X D (1) book value oday equals book value las year plus earnngs ( X ) less (ne) dvdends. The repored earnngs, dvdends, and book values do no always srcly adhere o he above cleansurplus relaon. In order o exacly sasfy hs assumpon n our sample, we defne our earnngs seres as he sum of dvdends and he change n book equy. Ths approach s parly dcaed by necessy (he early daa consss of book-equy seres bu does no conan earnngs). 5 Accounng and sock reurns reman o be defned. Le r denoe he log sock reurn and e he log clean-surplus accounng reurn on equy, defned as Subsung he log dvdend-growh rae, ME + D r log 1 + (2) ME 1 BE + D e log 1 + (3) BE 1 d, he log dvdend-prce rao, δ, and he log dvdend-obook-equy rao, γ d b, o he reurn defnons (2) and (3) yelds 6

9 r e = log(exp( ) + 1) + d + δ 1 δ (4) = log(exp( ) + 1) + d + γ 1 γ. (5) Fnally, we denoe he log BE/ME rao by θ : ( BE ME ) θ = log (6) An nconvenen nonlnear law descrbes he evoluon of a frm's BE/ME rao f ha frm pays dvdends. However, a lnear model can do a good ob of approxmang he nonlnear evoluon of θ = log ( BE ME ) e r = ρθ θ 1 + κ (7) where ρ s a parameer, κ approxmaon error, and ρ < 1 f he frms pay any dvdends and ρ = 1 f he frms do no. Ths approxmaon can be usfed by he followng logc. We approxmae boh sock and accounng reurns by a frs order Taylor seres approxmaon, choosng he same expanson pon: r = log(exp( ) + 1) + + δ 1 α ρδ + + δ 1 δ d d (8) e γ d d (9) = log(exp( ) + 1) + + γ 1 α ργ + + γ 1 Subracng (9) from (8) yelds he lnearzed accounng deny (7). Ierang (7) forward yelds he frs resul. 6 Usng he convenen lnear form of equaon (7) s possble o express he BE/ME rao as an nfne dscouned sum of fuure profably and sock reurns: = N N N N + r + e+ + 1 ρ ρ ρ κ + + ρ θ + N = = = θ 1, (1) If he BE/ME rao s well behaved and ρ < 1, he las erm of (1) converges o zero as N : θ 1 = ρ r + ρ e+ + ρ κ +. (11) = = = Ths approxmae model for a frm's BE/ME provdes he foundaon for our varance decomposon. B. Cross-seconal varance decomposon If all frms have saonary BE/ME raos wh he same uncondonal mean, he cross-seconal varance of frms' log BE/ME raos can be decomposed no wo pars: predcably of marke-adused sock reurns and predcably of marke-adused reurns on equy. In oher words, f here s crossseconal spread n log BE/ME raos, curren relave log BE/ME raos mus covary wh fuure relave log sock reurns and/or fuure relave log profably. In he secons ha follow, we wll ypcally refer o hese varables whou he relave, marke-adused, or log modfer for ease of descrpon. Smlarly, we 7

10 wll use uncondonal varance of BE/MEs o refer o he uncondonal varance of cross-seconally demeaned BE/MEs; he rue uncondonal varance would nclude me varaon n he cross-seconal mean of BE/MEs. To derve he varance decomposon, we mulply boh sdes of (12) by θ 1, drop he approxmaon error, and ake uncondonal expecaons: = [ r, θ ] ρ cov[ e, θ ] var( θ ) ρ cov (13) + 1 = Snce all frms are assumed o have saonary BE/ME raos wh equal uncondonal expecaons, he hrd erm, correspondng o he BE/ME rao far no he fuure, drops ou, even f ρ = 1. Equaon (13) shows ha (under he mananed assumpons) he uncondonal cross-seconal varance of frms BE/ME raos s due o cross-seconal covarance of fuure sock and/or accounng relave reurns wh pas BE/ME raos. Above, marke-adused quanes are denoed by ldes. I s mporan o noe ha var(θ ) n equaon (13) corresponds o he average squared marke-adused BE/ME rao, and ha hs varance merc s hus bes nerpreed as he ypcal cross-seconal dsperson n BE/MEs. Due o he nfne sums, mplemenng he above varance decomposon (13) requres an assumed auxlary sascal model and he assumpon of equal long-run BE/ME raos for all frms. If one s unwllng o make such assumpons, a dfferen varance decomposon can be used. The nfne sums n (13) can be replaced wh fne sums f an addonal erm s ncluded. Workng wh equaon (1) whou + 1 akng he lm N and repeang he above seps yelds var( θ ) N = ρ cov N [ 1 ] [ N + r, θ ρ cov e, θ ] ρ cov[ θ, θ ] = + N 1 (14) Compared o equaon (13), equaon (14) has an addonal erm: The las erm of (14) s a cachall predcably erm ha capures he profably and sock-reurn predcably beyond horzon N, as well as cross-seconal heerogeney of he mean BE/ME raos. A relave varance decomposon may be easer o nerpre. Assumng ha he cross-seconal varance of he BE/ME raos s no zero, one can dvde boh sdes of equaon (14) by he uncondonal BE/ME varance: 1 N = ρ cov var( θ ) N [ r, ] cov [ + θ 1 ρ e +, θ 1 ] = var( θ ) + ρ N + 1 [ θ, θ ] cov + N var( θ ) 1 (15) 8

11 The hree erms n (15) represen he fracon of varance arbuable o he hree sources. The relave varance decomposon s parcularly easy o nerpre each componen n (15) corresponds o a smple regresson coeffcen. The predcve regresson coeffcen of long-horzon reurns less he predcve regresson coeffcen of long-horzon profably plus a measure of he perssence of BE/ME spread mus be equal o one. Pas research ha decomposes he me-seres varance of aggregae porfolos ofen uses vecor auoregressons (VARs) nsead of long-horzon regressons. In our cross-seconal applcaon, he longhorzon regresson mehodology s preferable o he more common VAR mehodology. The dffculy wh he VAR mehodology arses from rebalancng effecs. Appendx 2 dscusses hs problem n more deal. C. Cross-seconal varance-decomposon resuls usng he U.S. panel To mplemen he cross-seconal varance decomposon, each year we creae 4 value-wegh porfolos of socks by sorng on BE/ME and rack he subsequen sock reurns, profably, and book-omarke raos of hese porfolos. For example, n 1997 (he las year of he U.S. sample) he low BE/ME porfolo has a value-wegh average BE/ME of.4, whle he hghes porfolo s a The mean porfolo BE/ME s.62, whle he sandard devaon across porfolos s.54. Dsperson n BE/ME has rsen and fallen many mes over he years. For 1987 he sandard devaon (.55) s almos dencal o he 1997 value, bu for 1991 s more han wce as hgh (1.18). Equaon (15) mples ha he coeffcens n he followng hree separae, forward-lookng regressons measure he percenage of nformaon n a frm s BE/ME rao concernng aspecs of he frm's fuure: = = ρ r ρ e N ρ θ +, + + = b( r, N) θ = b( e, N) θ = b( r, N) θ 1, 1, 1, + ε( r, N, ) + ε( e, N, ) + ε( θ, N, ) (16) We esmae hese coeffcens usng he 4 BE/ME-sored porfolos and repor he combned resuls n Table 1 Panel A. Thus, we regress he fuure reurns, profables, and book-o-marke raos for he 4 buy-and-hold porfolos on he porfolos curren book-o-marke raos. As we are neresed n a crossseconal varance decomposon, all varables are cross-seconally demeaned. For example, θ s he aggregae book equy of he socks n porfolo n he end of year dvded by he porfolo s aggregae, 9

12 marke value less he average of hese raos across he 4 porfolos n he end of year. In he sysem of regressons descrbed n equaon (16), b( r, N) b( e, N) + b( θ, N) 1. The N=1 row of he able breaks he BE/ME rao no nformaon abou fuure 1-year reurns, abou fuure 1-year ROE, and abou he fuure 1-year-ahead BE/ME raos. The spl s 3% for fuure reurns, 15% for fuure profably, and 83% for perssence n BE/ME raos. A one-year horzon, he BE/ME rao predcs all hree varables wh he expeced sgns. Because BE/ME raos are que perssen, he larges componen of he uncondonal varance of BE/ME raos s due o covarance wh nex year s BE/ME rao. Of course, nex year s BE/ME rao also has nformaon abou fuure reurns and ROEs beyond one year. Subsequen rows of he able explo hs by lookng furher ahead, o 2, 3, 5, 1 and 15 years. Fgure 1 graphs hese coeffcens as a funcon of he forecas horzon. 2% of BE/ME nformaon s abou 15- year reurns, 58% abou 15-year profables. The remanng 26% of he nformaon s abou 15-yearforward BE/ME raos, a resdual componen we wll refer o as he perssen componen of he BE/ME raos. The saonary and homogeney assumpons would guaranee ha even he perssen componen s nformave abou fuure reurns and profably beyond 15 years. Gven he facs abou he frs 15 years (.e., he reurn coeffcens level a 2% whle profably coeffcens connue o ncrease as a funcon of N), seems safe o assume ha mos of he remanng nformaon n he perssen componen of BE/ME raos concerns cash flows and no reurns. Our general nerpreaon of hese regresson resuls n Table 1 Panel A s hus as follows. Mos of he nformaon n he BE/ME raos of ndvdual frms s abou fuure profably. Abou 2% of crossseconal dfferences n BE/ME raos can be arbued o dfferences n expeced reurns. Dfferences n expecaons of fuure profably explan he oher 8%. Our procedure mposes he dscoun coeffcen rho (ρ) o be consan across frms. Because long-run dvdend yelds and pay-ou raos may vary sysemacally wh BE/ME raos, he assumpon of a consan dscoun coeffcen may lead o a poor approxmaon. Forunaely, our regressons provde a naural way o evaluae he effec of he approxmaon error on he varance-decomposon resuls. Repeang he dervaons n equaons (1)-(15) and carryng he approxmaon error along shows ha he varance componen due o he approxmaon error equals 1 b( r, N) + b( e, N) b( θ, N). The approxmaon error s share of he uncondonal varance s hus.1% a he one-year horzon and -4.74% a he 15-year 1

13 horzon. These compuaons ndcae ha he error n he consan-rho lnear approxmaon does no maerally affec our resuls. Table 1 Panel A also repors he sandard errors for he varance decomposon. Alhough we use regresson coeffcen pon esmaes obaned usng pooled OLS, he usual OLS sandard errors are lkely o be sgnfcanly undersaed. The problem wh OLS sandard errors arses from wo well-known sources. Frs, he regresson resduals may be correlaed n he cross-secon. If hs correlaon s relaed o he explanaory varables, he sandard errors are no vald. Second, because we use overlappng dependen varables, he regresson resduals are lkely o be auocorrelaed. A more general and precse saemen of hese problems s ha he covarance marx of he pooled regresson errors s no proporonal o an deny marx. In order o calculae approprae sandard errors ha accoun for correlaon of he resduals boh over me and n he cross-secon, we adap Rogers s (1983, 1993) sandard-error formulas o our regressons. Appendx 3 conans he deals of hese calculaons. No surprsngly, he sascal evdence ha he pas BE/ME raos predc fuure BE/MEs s overwhelmng, as -sascs n he las column are unformly hgh. Predcons of fuure BE/MEs sar ou a -sascs above 7 and reman above 2. even 15 years ou. There s a hgh degree of perssence n frms BE/ME raos. The ably of BE/ME o predc profably a longer horzons s also very convncng. The - sascs n he early years range from 3 o 11, and even oward he 15-year mark say above 2.. The reducon n sascal sgnfcance resuls from an ncrease n he sandard errors as we look farher no he fuure. We suspec ha hs s manly due o he longer predcon horzon generang a smaller number of ndependen observaons n our sample. Ths problem s parcularly severe n he predcon of reurns, especally for horzons longer han hree years. Alhough he coeffcens on fuure expeced reurns grow wh he horzon, he sandard errors grow even faser and as a consequence he -sascs fall below 2. n he laer years. Because we use longhorzon regressons and compue correced sandard errors, we canno reec he hypohess ha none of he BE/ME varance s due o expeced reurns a convenonal levels for he 15-year horzon specfcaon (he assocaed -sasc s 1.82). Ths s conssen wh our concluson ha, n he cross-secon, frms BE/ME raos are manly drven by expeced profably, no expeced reurns. 11

14 D. Cross-seconal varance-decomposon resuls usng he nernaonal panel Table 2 Panel A esmaes he sysem of regresson equaons (16) usng he nernaonal panel. Agan, he N=1 row of he able breaks he BE/ME rao no nformaon concernng he fuure 1-year reurns, fuure 1-year ROE, and fuure 1-year-ahead BE/ME raos. The spl s agan close o 3% for fuure reurns. In he nernaonal panel, he remanng poron conans less nformaon abou fuure profably (11%) and more abou perssence n BE/MEs (85%). As before, we look furher ahead. However, daa avalably lms our long-horzon esmaes o 2, 3, and 5 years. Afer 5 years, he precson of he varance-decomposon esmaes and he accuracy of he asympoc sandard errors decrease dramacally. 18% of BE/ME nformaon s abou 5-year reurns, 33% abou 5-year profables. The remanng 49% of he nformaon s abou he 5-year-forward BE/ME raos. One mgh argue ha a poron of he cross-seconal varance of he unadused BE/ME raos s due o permanen dfferences n accounng measures across counres. Ths argumen suggess ha once hs nose due o accounng sandards s removed, he counry-adused BE/ME should conan a hgher percenage of nformaon concernng reurns and/or ransory profably. Therefore, we refne our varance decomposon of Panel A by adusng each varable (BE/ME, reurn, and profably) by subracng he approprae value-wegh counry measure. However, he resuls (repored n Panel B) are que smlar o he unadused decomposon. In fac, for he adused daa, only 14% of he cross-seconal varance of counry-adused BE/ME s due o crossseconal dfferences n expeced reurns. As wh he unadused resuls, vrually half of he cross-seconal varance s due o perssence n counry-adused BE/ME. As menoned before, gven he saonary and homogeney assumpons, he perssen componen should be nformave abou fuure reurns and profably beyond 5 years. Ths fac combned wh our daa lmaons n he nernaonal panel make measuremen of he percenage of cross-seconal dfferences n he BE/ME raos arbuable o dfferences n expeced reurns more dffcul. Sll, seems safe o say ha, as n he domesc resuls, mos of he nformaon n he BE/ME raos of ndvdual frms s abou fuure profably. Whle hese nernaonal resuls are noser, we nerpre hem as supporng our man resuls obaned from he U.S. daa. 12

15 E. U.S. nra-ndusry resuls Ths secon examnes he role of ndusry n he peces of nformaon conaned n he BE/ME raos. We are neresed n wo ndusry-relaed phenomena. Frs, we decompose he cross-seconal varance of he ndusry-adused BE/ME raos. Ths decomposon answers he queson: o wha exen s whnndusry varaon of he BE/ME raos drven by whn-ndusry varaon n expeced profably and/or reurns? Second, we decompose he varance of raw (.e., no ndusry-adused) BE/MEs no sx componens by furher separang he profably, reurn, and BE/ME-perssence componens of Table 1 Panel A no ner- and nra-ndusry pars. Ths decomposon shows wheher ner-ndusry or nrandusry phenomena are he man drvers of he cross-secon of raw BE/MEs. Table 1 Panel B decomposes he varance of nra-ndusry BE/MEs usng Fama-French (1996) ndusry classfcaons. Jus as perssen dfferences n accounng sandards across counres mgh nroduce nose o comparsons of BE/ME among socks of dfferen naons, smlarly perssen dfferences n accounng sandards across U.S. ndusres could cause perssen BE/ME dfferences. Such dfferences n ndusry BE/ME would be unrelaed o near-erm predcably of eher profably or reurns. In secon II.C s decomposon, (Table 1 Panel A) we sor frms no 4 porfolos based on her raw BE/ME rao. Bu n Panel B we sor socks no porfolos based on nra-ndusry BE/ME. Inra-ndusry BE/ME s defned as he dfference beween a frm s BE/ME and he value-wegh average BE/ME of he ndusry he frm s n. Thus he es asses for Table 1 Panel B are porfolos ha are more ndusry-balanced han hose n Table 1 Panel A as every ndusry s lkely o have frms ha are hgh, medum, and low n her ndusry-relave BE/ME rao. We fnd ha he mporance of fuure reurn, ROE, and BE/ME n explanng curren BE/ME relave o a frm s ndusry, s que comparable o her mporance n explanng overall BE/ME. Over 15 years, reurns explan 19% of cross-seconal varaon, ROEs 58%, and fuure BE/ME 27%, smlar o he correspondng numbers n Table 1 Panel A. Table 3 examnes he relave mporance of ndusry and nra-ndusry varaon o he BE/ME crosssecon. The es porfolos are creaed by sorng on he raw BE/ME raos, and he only dfference beween he porfolos used n Table 1 Panel A and Table 3 s ha porfolos used n Table 3 exclude socks ha canno be assgned o an ndusry of a leas en frms. In an exenson of he concep of nra-ndusry BE/ME, each of he hree varance componens fuure reurns, fuure ROEs, and fuure BE/ME are 13

16 subdvded no wo peces. The frs peces, labeled nra-ndusry n he able, are compued for each frm as he excess of s reurn, ROE or fuure BE/ME over he reurn, ROE or fuure BE/ME for he ndusry he frm s n. The second componen, labeled ndusry n he Table, s smply he value-wegh reurn, ROE, or fuure BE/ME for he relevan ndusry as a whole. Snce a frm s reurn n any gven year s he sum of he reurn on he frm s ndusry for ha year and he excess reurn of he frm over ha ndusry reurn; and snce he same can be sad for ROE or fuure BE/ME, s clear ha we can replace our 3-componen decomposon wh a new 6-componen decomposon, whle preservng he orgnal deny. The coeffcens n Table 3 show ha regardless of horzon roughly 8% of he nformaon n he raw BE/ME rao concerns he frm s behavor relave o s ndusry, whle he remanng percenage s nformave abou he ndusry as a whole. Lookng a sngle year ahead, he coeffcen on nra-ndusry reurn s.21, whle ha on ndusry reurn s.55. The ROE coeffcens are and -.14, respecvely, whle he BE/ME coeffcens are.634 and.194, respecvely. Ths suggess ha whle he maory of he reurn nformaon n he BE/ME rao s abou nra-ndusry reurn, a non-rval poron concerns he reurn on he ndusry as a whole. Ffeen years ou, he nra-ndusry reurn explans 16.4% of frm BE/ME, whle ndusry reurn explans 4.5%. These resuls are smlar n spr o Lewellen (1999). Hs analyss documens ha conrollng for ndusry does no sgnfcanly reduce he degree of cross-seconal spread n sensvy o he HML facor of Fama and French (1993). The domnance of nra-ndusry nformaon s even greaer n he profably seres. The coeffcen on nra-ndusry ROE s nne mes larger han ha on ndusry ROE afer one year, and s nneeen mes larger (-.53 vs. -.26) afer ffeen years. I s no surprse ha mos of he nformaon n he BE/ME raos s abou nra-ndusry performance raher han ndusry performance afer all, f hs were no he case, BE/ME would essenally us be a proxy for ndusry. Bu s neresng o observe ha as regards he relaon beween BE/ME and reurns, boh nra-ndusry and ndusry componens are mporan, whle he relaon beween he BE/ME rao and fuure ROE s prmarly based on ndusry-relave nformaon. F. Does he varance-decomposon vary as a funcon of frm characerscs The relave mporance of he hree elemens of he decomposon -- ransory varaon n expeced reurns, ransory varaon n profably and perssan dfferences n he BE/ME raos -- may be dfferen for dfferen frms. Such varaon can be examned va a condonal varance decomposon. We analyze hs possbly by nally sorng frms each year no hree groups based on a parcular frm characersc. 14

17 We hen sor frms no fve porfolos based on BE/ME and esmae he varance decomposon separaely whn each of he hree groups. Ths condonal varance decomposon s smply he bes esmae of equaon (15) for a parcular ype of frm. Naural frm characerscs o examne are sze and he BE/ME rao self, whch have he advanage ha condonng on hem does no reduce he sze of he sample. Sze (.e. marke capalzaon) may proxy for lqudy or speed of nformaon dsperson, facors whch may nfluence he amoun of ransory varaon n reurns. Smlarly, a varance decomposon condonal on he BE/ME rao may hghlgh asymmeres n he amoun of ransory varaon due o expeced reurns, perhaps as a resul of shor sale consrans ha preven lqudy provders from sellng socks wh very low expeced reurns. Therefore, we sor frms no hree sze groups based on NYSE breakpons o examne how he varance decomposon vares wh marke capalzaon (Table 4 Panel A) and we sor frms no hree groups based on frm BE/ME (Table 4 Panel B) o examne how he decomposon depends on BE/ME. The decomposon condonal on sze ndcaes ha he book-o-marke raos have less nformaon concernng dfferences n expeced reurns among large frms han small or medum-szed frms. However he dfferences do no seem economcally large and are no sascally sgnfcan. For large frms, approxmaely 16% of he nformaon n he BE/ME raos s due o dfferences n expeced reurns whle for small frms hs number s 18%. One mgh nally be surprsed by hs resul, as prevous leraure, for example Fama and French (1993), has generally ndcaed ha he value effec s sronger among small socks. These wo poenally conflcng peces of evdence are reconclable. Frs, he spl s que dfferen a shor horzons (4.5% due o expeced 1-year reurns for small frms and only 2.3% for large). However, hs dfference s counerbalanced by he fac ha a shor horzons, large socks have more perssen BE/ME. The regresson coeffcen of fuure 1-year BE/ME on curren BE.ME s.8757 for large socks bu only.8141 for small socks. Small sock BE/MEs have more reurn nformaon n he shor run, bu also more profably nformaon. In he long run hese effecvely cancel, leavng he oal spl beween reurn and profably nformaon smlar for small and large socks. Second, noe ha snce N- year reurns n he frs equaon n he sysem of regressons n (16) are gven by he produc, b( r, N) θ 1, a large cross-seconal varaon n expeced reurns can generaed by eher a large b ( r, N) or a large spread n θ 1. Ths observaon s cenral o our ably o forecas he reurns on value versus growh sraeges n secon III and apples here as well. Small socks do have a sronger value effec, even a long 15

18 horzons, no because her BE/ME raos have more nformaon concernng fuure reurns bu raher because her BE/ME s are more dspersed. Ths dfference n dsperson s subsanal: For he smalles hrd, he average cross-seconal varance of log BE/ME for he fve BE/ME qunles was.53; hgher han he correspondng value of.39 for he larges hrd of he sample. Panel B of Table 4 allows he decomposon o vary wh BE/ME. Largely by consrucon, he medum-be/me hrd has an average cross-seconal varance of log BE/ME of us.3, whle boh he hgh-be/me (.14) and low-be/me (.22) socks exhb far more spread. A he 1-year horzon, hgh- BE/ME frms BE/MEs have more nformaon concernng expeced reurns. As he horzon ncreases, he percenage of nformaon n he BE/ME raos concernng dfferences n fuure reurns remans relavely low for low BE/ME frms. I s only a he 15-year horzon ha he nformaon conen becomes roughly equal. In general, he dfferences we documen n he decomposon as a funcon of sze and BE/ME are no economcally large. Moreover, our general concluson ha mos of he cross-seconal dsperson n he BE/ME raos s due o dfferences n cash flows s conssenly rue across he subses of frms we sudy. G. Does he varance decomposon vary as a funcon of marke-wde nsrumens? The relave mporance of he hree drvers of he value spread may also vary over me. Ths me varaon can also be examned va a condonal varance decomposon. In hs case, he condonal varance decomposon s smply he bes esmae of equaon (15) a any pon n me. Whle here are many ways o esmae a condonal verson of (15), we begn wh perhaps he smples approach. We wre he hree regresson coeffcens n (15) as lnear funcons of varables wh nuve predcve conen. We lm ourselves o smple marke-wde nsrumens ncludng he medan frm's BE/ME rao, he cross-seconal varance of frms' BE/ME raos, he cross-seconal varance of frms' profably, he crossseconal covarance of frms' BE/ME raos and profably, and bond-yeld varables. Our choce of marke-wde nsrumens aemps o use nformaon n a parcular cross-secon of he BE/ME raos o draw nferences abou he me-seres properes of he nformaon n a ypcal frm's BE/ME rao. For example, f he correlaon beween he BE/ME raos and frm profably s hgher han normal, hen one mgh expec ha he ypcal frm's BE/ME rao probably conans more nformaon concernng fuure expeced cash flows han fuure expeced reurns. Smlarly, f he ypcal frm s cheap as represened by a 16

19 hgher han normal medan BE/ME, one mgh expec he ypcal frm's BE/ME rao o conan more nformaon concernng expeced reurns. Ths concluson follows from he resuls of Vuoleenaho (2) who fnds ha mos of he me varaon n he aggregae BE/ME rao s due o changes n fuure expeced reurns. We do no show he resuls of hese condonal varance decomposons because, n general, we do no fnd any perods n me where BE/ME raos conan more nformaon abou expeced reurns. Thus he percenage of nformaon n he BE/ME rao s relavely consan, perhaps surprsngly so. Of course, as we emphaszed earler, he value spread does move que a lo hrough me. If he nformaon conen n he value spread s consan, hen expeced reurns o a value sraegy should exhb rch, me-varyng paerns as he value spread moves around. We analyze hs possbly n he nex secon. III. Predcng value versus growh reurns Consder a porfolo ha s long hgh-be/me socks and shor low-be/me socks. Fama and French (1993,1996) popularze such a zero-nvesmen porfolo (HML) n numerous applcaons. Frs, Fama and French consruc sx value-wegh porfolos from he nersecons of he wo sze and he hree BE/ME groups. The HML porfolo s hen formed by buyng boh he small and he large hgh-be/me porfolos (combned poson denoed by H) and sellng shor boh he small and he large low-be/me porfolos (combned poson denoed by L). The wo componens of HML are hus hgh- and low-be/me porfolos wh abou he same weghed-average sze. The condonal varance decomposon above can be used o movae a forecasng model for he reurn on he HML porfolo. Apply (12) o boh he H and L porfolo, dfference, and reorganze: = ρ E H L H = ( θ 1 θ 1 + ρ E 1e + ρ E = = HML 1 r + ) 1 e L + (17) Ths movaes a predcve regresson: R HML H L H L = a + b ( θ 1 θ 1 ) + c ( e 1 e 1) + ε, (18) where he pas profably spread s used as a proxy for he spread n dscouned fuure expeced profably. Snce mos of he emprcal work ha reles on he me-seres of reurns on he HML porfolo uses smple (no log) reurns, we use annual smple reurns as he dependen varable n our regresson. 17

20 We presen OLS esmaes of he coeffcens n HML forecasng regressons smlar o equaon (18). Furhermore, as one mgh expec an ncrease n he expeced reurn on HML o be assocaed wh an ncrease n he volaly of he HML reurn, we also produce maxmum-lkelhood GLS esmaes based on an accompanyng model (usng he same nsrumens) of he log varance of he HML porfolo: HML R Z β ε ε N[, exp( Z )], (19) = 1 +, 1γ where Z 1 are he lagged predcor varables (ncludng he value spread) and β, γ are parameers. Snce he above specfcaon produces condonal varance esmaes, enables us o compue a me-seres of esmaed Sharpe raos as well as esmaed expeced reurns. For a dealed exposon of he erave esmaon procedure and he sandard-error formulas, see Greene (1997, p ). Fgure 2 dsplays he log BE/ME of he hgh-, medum-, and low-be/me porfolos creaed by Fama and French (1993). One can see from he fgure ha he log BE/MEs of he hree sze-balanced porfolos move around que a b over he 61-year perod. Whle he level of BE/MEs appears volale and perssen, he value spread (he dfference beween H and L porfolos BE/MEs) appears o be followng a mean reverng process. Table 5 Panel A shows he regresson resuls. In ha able, we repor he coeffcens n an OLS regresson, he GLS counerpar, and he coeffcens ( γ ) n he exponenal-lnear condonal-varance model. In he dscusson, we focus on he GLS esmaes of he condonal mean. As expeced, we fnd ha he dfference beween BE/MEs of he low- and hgh-be/me porfolos, he value spread, s a sgnfcan predcor of he reurn on he HML porfolo. The smple regresson coeffcen of he value spread s.287; he -sasc s Ths resul ndcaes ha he annual expeced reurn on HML s mevaryng. As he annual sandard devaon of he value spread s 8.75 percenage pons, hs predcve regresson mples subsanal me varaon n he HML premum: he sandard devaon of he fed values of HML s 1.3 mes he uncondonal mean HML reurn. Fgure 3 graphs he expeced reurn on he HML porfolo usng hs specfcaon. As a measure of he economc sgnfcance of our fndng, Fgure 3 also graphs he assocaed condonal Sharpe rao. As specfed n equaon (19), we generae hese esmaes each year by usng he value spread o predc he reurn and log varance of HML. In general, we fnd ha he pon esmaes of he HML Sharpe rao vary over me consderably. 18

21 In order o examne he robusness of our resuls o daa-snoopng concerns, we re-esmae hs smple forecasng model usng he nernaonal panel. We hope ha he nernaonal sample provdes an ou-of-sample es of our fndng of me-varaon n he expeced reurn on value-versus-growh sraeges n he US sample. As wh he US value spread, our forecasng varable seems well-behaved. As he nernaonal sample s only 17 years long, we esmae he predcably usng counry-adused varables n he hope of ncreasng he precson of our regresson coeffcens. The resuls, repored n Table 5 Panel B, are mprecse bu he coeffcens are conssen wh he domesc esmaes. Table 5 Panel A also conans mulple regressons wh addonal predcor varables. In he mulple regresson specfed by equaon (18), he value spread s coeffcen and -sasc are.2915 and 3.14, respecvely. However, he recen dfference n ROE s beween value and growh socks does no provde addonal predcve power (coeffcen of.3, -sasc.5913). We also repor a specfcaon wh he value spread neraced wh he medan BE/ME rao of he marke. In he condonal varance decomposons dscussed n secon II.F, hs varable, hough sascally nsgnfcan, was he mos successful n rackng me-varaon n he nformaon conen of frms BE/ME rao. As wh he condonal varance decomposons, hs varable s only margnally sgnfcan. We complee our analyss of me-varaon n he reurn on value-versus-growh sraeges by nvesgang wha macroeconomc varables are correlaed wh he value spread. In parcular, we regress he value spread on several varables ha nuvely mgh explan movemens n he value spread. These regressons are shown n Table 6. We nally regress he value spread on he medan BE/ME rao. Ths varable has no explanaory power by self (-sasc of.12). We urn o he defaul spread (he dfference n yelds beween BAA and AAA long-erm corporae bonds) ha, lke he medan BE/ME rao s an ndcaor of low frequency movemens n busness condons. We fnd ha he value spread has a posve and sascally sgnfcan coeffcen on he defaul yeld spread. The coeffcen s.153 wh an assocaed -sasc of 2.. Almos a quarer of he varaon n he value spread can be lnked o movemens n hs varable alone. We also nclude he medan BE/ME rao and he lagged profably spread: The medan BE/ME rao s now sascally and economcally sgnfcan and he lagged profably spread s margnally so. The full model now explans over 42% of he movemens n he value spread. 19

22 When we add boh he medan BE/ME rao and he defaul spread o he reurn-predcve regressons n Table 4, we fnd ha he value spread remans sgnfcan (coeffcen of.394, -sasc of 3.1) whle he medan BE/ME rao s now sascally sgnfcan (coeffcen of.128, -sasc of 1.8) a he 1% level. Unrepored orhogonalzed regressons show ha he componen of he value spread ha s correlaed wh he marke s BE/ME rao or defaul yeld spread does no predc HML reurns. The predcve ably of he value spread s enrely due o he componen ha s orhogonal o hese markewde nsrumens. IV. Conclusons The presen-value formula allows us o decompose he cross-seconal varance of frms book-omarke raos no hree componens: 1) covarance of fuure sock reurns wh he pas book-o-marke raos, 2) covarance of fuure profably (.e., accounng reurn on equy) wh he pas book-o-marke raos, and 3) perssence of he book-o-marke raos. We esmae hs decomposon from a large ( ) panel wh hree smple long-horzon regressons. Our resuls sugges ha approxmaely 2% of he cross-seconal dsperson of book-o-marke raos can be explaned wh expeced 15-year sock reurns, 58% wh expeced 15-year profably, and 26% wh 15-year perssence of book-o-marke raos. Inuon and he me-seres behavor of reurns and profably sugges ha he perssence of he book-o-marke raos s mosly due o cross-seconal varaon n expeced profably beyond he 15-year horzon. Hence, we aggressvely nerpre our regressons as suggesng ha approxmaely 2% of he dsperson n he book-o-marke raos s due o dsperson n expeced sock reurns and 8% due o dsperson n expeced profably. We documen smlar resuls for an nernaonal panel coverng 23 counres (excludng he US) over he perod. As wh he domesc panel, mos of he varaon n he book-o-marke raos, even afer adusng for dfferences n accounng pracces across counres, s due o nformaon concernng expeced fuure profably. One could conecure from he above varance decomposon resul ha he expeced annual premum on Fama and French s (1993) HML porfolo s me-varyng. Our emprcal evdence confrms ha supposon: he expeced reurn on a value-mnus-growh sraegy s aypcally hgh a mes when he value spread s wde and he marke s cheap. 2

23 Appendx 1: U.S. daa In order o be ncluded n our U.S. sample, a frm-year mus sasfy he followng daa requremens. When usng COMPUSTAT as our source of accounng nformaon, we requre ha he frm mus be on COMPUSTAT for wo years. Ths requremen avods poenal survvor bas due o COMPUSTAT backfllng daa. Also, due o daa-qualy concerns, all predcve ess requre he dependen varable o correspond o year 1937 or laer. Book equy s defned as he sockholders equy, plus balance shee deferred axes (daa em 74) and nvesmen ax cred (daa em 28) (f avalable), plus pos-reremen benef lables (daa em 33) (f avalable) mnus he book value of preferred sock. Dependng on avalably, we use redempon (daa em 56), lqudaon (daa em 1), or par value (daa em 13) (n ha order) for he book value of preferred sock. Sockholders equy used n he above formula s calculaed as follows. We prefer he sockholders equy number repored by Moody s, or COMPUSTAT (daa em 216). If neher one s avalable, we measure sockholders equy as he book value of common equy (daa em 6) plus he par value of preferred sock. (Noe ha he preferred sock s added a hs sage because s laer subraced n he book equy formula.) If common equy s no avalable, we compue sockholders equy as he book value of asses (daa em 6) mnus oal lables (daa em 181), all from COMPUSTAT. The BE/ME rao used o form porfolos n May of year s book common equy for he fscal year endng n calendar year -1, dvded by marke equy a he end of May of year. 7 We requre he frm o have a vald pas BE/ME. Moreover, n order o elmnae lkely daa errors, we dscard hose frms wh BE/ME raos less han.1 and greaer han 1. When compung sock reurns, we nclude delsng daa when avalable on he CRSP apes. In some cases, CRSP records delsng prces several monhs afer he secury ceases radng and hus afer a perod of mssng reurns. In hese cases, we calculae he oal reurn from he las avalable prce o he delsng prce and pro-rae hs reurn over he nervenng monhs. The clean-surplus ROE s calculaed as follows. We compue he frm's annual earnngs usng he assumpon of clean surplus accounng and he frm's dvdends from CRSP. The formula used for compung log ROE s (1 + R ) M 1 D B = D e log 1, (A2.1) M B 1 B 1 21

24 where M denoes marke and B book equy, D dvdends, and R he sock reurn. Ths relaon s smply ha earnngs hs year equals he change n book equy plus dvdends wh an approprae adusmen for equy offerngs. Appendx 2: Why no VARs wh managed porfolos? A frs, may seem ha a VAR model would be a smpler and more elegan alernave o our longhorzon regressons. I s empng o nclude, for example, HML reurn, HML value spread, and HML profably spread no a VAR model sae vecor and compue he varance decomposon along he lnes of Campbell and Shller (1988) and Vuoleenaho (2). I urns ou, however, ha he economc nerpreaon of he VAR-based varance decomposon may be maerally dfferen from he long-horzon regresson varance decomposon we advocae. The dfference beween he wo mehods orgnaes from he fac ha he HML porfolo s weghs are managed. 1 θ, and To llusrae hs pon, consder a general managed porfolo seres and a VAR model. Le 1 r, 1 e denoe he log reurn, log book-o-marke, and log profably on a buy-and-hold porfolo. The superscrp n he above varables denoes he me when he buy-and-hold porfolo was formed. Adapng he basc lnearzed book-o-marke law (7) o hs noaon yelds: e r ρθ θ (A1.1) As one can see, he basc deny descrbes he evoluon of a buy-and-hold porfolo s book-o-marke. An equaon descrbng he book-o-marke evoluon of a managed porfolo mus nclude an addonal erm due o rebalancng. Addng ρ ( θ 1 θ ) o boh sdes of (A1.1) resuls: e 1 r 1 + ρ ( θ θ 1 ) ρθ θ 1 1 (A1.2) Equaon (A1.2) descrbes he evoluon of a managed porfolo seres. The change n managed porfolo book-o-marke s explaned by hree erms: managed porfolo profably, managed porfolo reurn, and a rebalancng erm. Comparson of equaons (A1.1) and (A1.2) hghlghs he dfference beween a VAR varance decomposon and a long-horzon regresson varance decomposon. A long-horzon regresson s based 2 1 on eraon of (A1.1). A VAR-model maps a vecor [ r, θ, e ] o [ r, θ, 1 e ] and, hus, can only mplemen a varance decomposon based on erang equaon (A1.2). The more aggressvely he porfolo s rebalanced, he larger he dfference beween he wo. 22