THE VOLATILITY OF THE FIRM S ASSETS

Transcription

1 TH VOLTILITY OF TH FIRM S SSTS By Jaewon Cho* and Mahew Rchardson** bsrac: Ths paper nvesgaes he condonal volaly of he frm s asses n conras o exsng sudes ha focus prmarly on equy volaly. Usng a novel daase ha allows us o map ou sgnfcan porons of he capal srucure, we examne he volaly properes of asse reurns as calculaed by a weghed average of equy, bond and loan prces. The wo fundamenal fndngs n hs paper are ha asse volaly s me-varyng and ha fnancal leverage maers and has a large nfluence on equy volaly. Whn hs backdrop, several new resuls emerge. Frs, leverage plays a more mporan role han prevously hough n explanng he well-documened asymmerc volaly effec. Second, equy volaly possesses boh a ransory componen due prmarly o asse volaly and a more permanen componen due o fnancal leverage. Thrd, n erms of a breakdown of he deermnans of equy volaly, we relae mpled equy volaly levels and changes o dfferen componens of esmaed asse volaly (.e., boh dosyncrac and marke, ncludng lagged volaly and asymmerc reurn shocks) and o leverage a he frm level. *Sern School of Busness, NYU ** NBR and Sern School of Busness, NYU We would lke o hank Rob ngle and semnar parcpans a he Inaugural Conference of SoFI.

2 I. Inroducon Undersandng why asse (.e., frm value) volaly changes hrough me s a fundamenal ssue n fnance. Ths s because asse volaly plays a key role boh n deermnng capal srucure valuaon and he sandard reurn/rsk radeoff ndependen of fnancal leverage. Surprsngly, very lle, however, s known abou he cross-seconal and me-seres properes of asse volaly. Due o he lack of comprehensve daa on publc deb, he focus of he fnance leraure has been o analyze equy reurn volaly wh occasonal references o he leverage effec. Ths paper provdes a dealed examnaon of asse volaly across a broad crosssecon of publcly raded frms usng a novel daase ha ncludes prces and oher nformaon on eques, publcly raded deb and syndcaed loans. Ths daase allows us o map ou sgnfcan porons of he frm s capal srucure. Vewng he frm s asses as a porfolo of he ndvdual secures whn he frm (a la Modglan and Mller (96)), we are able o esmae he reurn on he frm s asses from a weghed-average reurn on hese ndvdual componens. Measuremen error asde, hs provdes a dsnc advanage o he exsng leraure. Specfcally, we can drecly address quesons relang o asse volaly whereas prevously hey were mpled from a jon analyss of equy volaly usng lmed daa on a frm s deb and overly smple models of a frm s capal srucure. Because hese models are mos probably no accurae descrpons of realy, he lnk beween equy and asse volaly s broken. s jus one llusraon, one of he mplcaons of hese models s ha he ndvdual secures may have nonsaonary and complex, nonlnear forms for expeced reurns as rsk ges shfed across secury classes whn he frm (as he probably of dsress moves around). The underlyng asse reurn (.e., he porfolo of hese secures), however, s more lkely o be beer behaved and more conducve for sandard emprcal analyss. e.g., see Black and Scholes (973), Meron (974), and he prolferaon n cred marke research over he las decade, for example, Black and Cox(976), Leland and Tof(996), Longsaff and Schwarz(995), Colln-Dufresne and Goldsen(00) e.g., French, Schwer and Sambaugh (987), Glosen, Jagannahan, and Runkle(993), Whelaw (994), Leau and Ludvgson(003), Brand and Kang(004), Ghysels, Sana-Clara and Valkanov (005), Bal and Peng (004), and Guo and Whelaw (006)

3 s an applcaon, we are able o provde new evdence on a heavly researched area, namely he sylzed fac ha sock reurn volaly rses afer sock prces fall. There has been consderable debae abou how much of hs effec s due o fnancal leverage as a resul of he sock prce fall (.e., leverage effec ) versus me-varyng rsk prema (.e., volaly feedback ). (See, for example, Black (976), Chrse (98), French, Schwer, and Sambaugh (987), Campbell and Henschel (99), ngle and Ng (993), Duffee (995), Bekaer and Wu (000) and Wu (00), among ohers). Moreover, now added o he fray are behavoral economss who argue ha hs sylzed fac may be due o nose radng on he par of rraonal agens. (For some recen dscusson of hs leraure wh applcaon o dosyncrac volaly, see, for example, Chen, Hong and Sen (00), Goyal and Sana Clara (003), and Hong and Sen (003)). We provde an esmaed breakdown of how much of a frm s equy volaly s due o he varous componens, such as fnancal leverage, rsk prema, me-varyng asse volaly, and so forh. The man concluson s ha he level of condonal equy volaly of a frm s mosly descrbed by fnancal leverage, he lagged asse volaly of markes, and he lagged asse volaly of he frm. In conras, changes n hs equy volaly are explaned by fnancal leverage, asymmerc shocks descrbed by he curren sock marke reurn (.e., rsk prema effec) and he frm s asse reurn (.e., dosyncrac rsk effec), and no mean-reverson n volaly. Ths paper provdes several addonal conrbuons o our exsng emprcal knowledge of frm volaly. Frs, n erms of new sylzed facs, we documen a very srong negave relaon beween asse volaly and leverage. Ths s a poenally mporan fndng because suggess one may have o be careful of lookng a equy volaly and leverage ogeher. In oher words, hey are jonly deermned by he volaly of he frm s asses as many corporae fnance heores would ex ane sugges. Second, whle here s clear evdence of he exsence of a leverage effec, mos of he explaned varaon of volaly can be arbued o me-varaon n he underlyng dosyncrac asses of he frm. Ths evdence s confrmed boh a he porfolo and ndvdual frm level and poses some challenges for fuure research. Thrd, conssen wh he leraure ha documens boh ransory and permanen componens o condonal equy volaly, we show ha he ransory componen s due prmarly o asse volaly and he more permanen componen s due o fnancal leverage.

4 Ths paper s organzed as follows. Secon provdes a dealed analyss of he daa. In parcular, we descrbe how each frm s capal srucure s mapped ou gven he varous daa sources. Specal aenon s devoed o he saleness of he daa. Some mporan and new sylzed facs are provded. Secon 3 presens he me-varyng properes of volaly boh a he ndvdual, ndusry and ndvdual frm level. s an applcaon, n Secon 4, we esmae he conrbuon of varous proposed sources of volaly o a frm s equy volaly. Secon 5 concludes. II. Daa Descrpon In order o map ou he capal srucure and consruc he reurns on a frm s asses, we need o ulze a number of daases, ncludng () CRSP for equy prces, () he Brdge JV daabase from Reuers for corporae bond prces and deals, () he FISD from Mergen for addonal corporae bond deals and checkng of he JV daa, (v) Dealscan and he mark-o-marke prcng servces from Loan prcng Corporaon for loans, (v) Compusa for he face value of deb and oher accounng nformaon, and (v) Bloomberg for fac checkng dscrepances. The consrucon of he asse reurn seres and he descrpon of he daa are provded n deal n Cho (008). s a resul, we summarze he less well-known daa and he asse reurn consrucon brefly below, and hen provde some sylzed facs.. Daa Sources The mos mporan daa source n hs paper are he corporae bond prces gven n he JV daabase of Reuers. ach day, he bd and ask prces are gahered from dealers n he markeplace and hen aggregaed o one se of bd and ask prces. s an ndcaon of s mporance n he corporae bond marke, mos parcpans use hs daabase o mark her books each day. In erms of he sample perod, he daabase covers he perod from July 99 o December 007, alhough he daa s que spoy pror o he md 990s. The bond daa requres subsanal cleanng. For example, a number of bonds are ssued under rule 44a before laer beng exchanged o he publc marke. Ths creaes numerous perods of double counng n he daase. In addon, bonds are called, convered and endered whch also leads o errors. To he exen possble, he daa s carefully mached 3

5 agans he Mergen Fxed Income Secures Daabase (FISD) and, when approprae, hand checked agans daa provded n Bloomberg and 0-K flngs (especally around frms showng a change n her ousandng deb). We seleced bond ssuers ha have CRSP sock reurn and Compusa accounng nformaon, and ha were nonfnancal n naure (due o he degree of leverage and wha ha means for fnancal frms). 3 The wo mos serous problems wh he bond prce daa are he poenal for saleness and marx prcng. Wh respec o saleness, many of he bonds do no rade on a frequen bass, so he daly quoes reflec average bd prces of he dealers for unraded secures. Because hese bds are only ndcave n he markeplace, updang s perhaps less mporan han oher secury markes. Ths pon asde, hese updaes may reflec sloppy marx prcng, leadng o excessve comovemen whn a frm s secures and possbly across smlar ypes of frms. Secon II.C below wll provde descrpve sascs on some of hese ssues. Neverheless, o allevae he problem, we () use monhly daa closes o he end of monh as hese prces end o be more carefully updaes (Warga (99)), and () only look a frms wh a leas $50mm of oal asses. Ths laer resrcon avods levered frms wh small bond ssues ha rarely rade. Table summarzes he coverage of our sample relave o he usual CRSP/Compusa unverse. I also shows he effec of droppng fnancal frms and frms wh a low amoun of marke value of her asses. The able repors several summary sascs. Table shows ha he prmary dfference beween he wo samples s one of asse sze. For example, resrcng he comparson o frms wh deb, he medan sze of he frm s marke value of asses s $.9 bllon for our unverse versus $0.3 bllon for he CRSP unverse. Whle hs fac s margnally relaed o our resrcng he sample o $0.5 bllon sze frms, he prmary reason s ha he bond daa base does no nclude small frms wh small amouns of deb. In parcular, he medan marke value of asses/equy rao (.e., leverage rao) s.5 versus.5 n he wo samples. frs glance, hs mgh sugges ha our bond unverse s small. Ths s no he case, however, as we overlap wh over 90% of he bonds n he Mergen FISD. Of some noe, Table B shows he leverage whn our sample across rangs classes, namely wh medan leverage rao of (.05,.6,.3,.48,.65,.96, and 3.4) for (,,, BBB, BB, B and CCC), respecvely. 3 Frms wh subsanal fnancal operaons, such as General lecrc, General Moors and Ford, were also 4

6 More mporanly, n erms of machng our frms o hose n Compusa, he publc bonds n our sample cover on average 56% of Compusa s long-erm deb frm by frm. Table B looks a hs calculaon more closely by calculang he coverage across frms n dfferen rangs classes. The able suggess he coverage s much hgher for hgh-yeld frms han for nvesmen grade frms. For example, he and frms percenages are 4.8% and 43.7%, respecvely whereas he B and CCC (and lower) frms have 73.% and 68.6% coverage, respecvely (For a more dealed descrpon of he characerscs of he daa, see Cho (008).) s shown n Table, only a poron of he deb of a company comes n he form of publcly raded bonds. consderable poron can be explaned by bank loans. The major sources for he bank loan daa are Dealscan (gong back o 987), and, for he prcng and more dealed characerscs of he loans, he Loan Syndcaons and Tradng ssocaons (LST) and Loan prcng Corporaon (LPC). There have been some analyses of he qualy of he prcng daa, mos noably Taylor and Sansone (007). The man concluson ha, a leas for cases where raded prces are avalable, he average dealer marks are represenave. One drawback of he daa s () ha s avalable over a much shorer me perod, and () ha acve volume, and hus relable secondary prces, occur only for leveraged loans. Of course, bank loans of nvesmen grade frms end o rade around par f her coupon raes floa. For he concden perod n whch we have access o boh bond and loan daa, Table shows ha over 94% of he capal srucure s covered. B. Consrucon of sse Reurns ssumng Modglan and Mller (96), he frm s asses and lables exacly offse, so ha we can represen he reurn on a frm s asses by s weghed average reurn of s underlyng fnancal clams. In order o calculae hese reurns, we herefore need o () map ou he frm s enre capal srucure (and s correspondng secures), and () record prces and nerm paymens of each pece of he capal srucure. The capal srucure for each frm s mapped ou monh by monh usng all he above daases. Because of he dynamc naure of he frm s capal srucure, n parcular, he deb amoun ousandng changes for a number of reasons, he daases are no always excluded. 5

7 algned. 4 s menoned n II.. above, dscrepances n amouns ousandng or oher dfferences n he daa were generally refereed manually usng Bloomberg s corporae acons em or 0-K flngs. For he perod whch we have boh bond and loan daa, Table I shows ha we can denfy mos of he capal srucure of he frm. For perods n whch we have only bond daa, we feel comforable herefore assumng he dfference beween longerm deb and he publc deb are loans. Gven hs mappng, how do we measure he reurns on he asses of he frm? ppealng o Modglan and Mller (96), he value of he real asses can be represened by he value of he fnancal asses, so ha wo dencal frms wh que dfferen capal srucures can have he same value of s underlyng asses. Ths allows us o wre he reurn on he asses of he frm as a weghed average of he reurn on each of he frm s fnancal asses, he weghs beng deermned by he relave value of each of he fnancal asses. In erms of each ndvdual componen, equy reurns are calculaed he usual way from monh o monh, as nex perod s prce plus any dvdends pad dvded by he curren prce. Bond reurns are calculaed smlarly each perod from he quoed bond prces, coupon and accrued neres. 5 The more rcky calculaon revolves around he reurns of bank loans. On he posve sde, because bank loans resde owards he op of he capal srucure (or a leas unl que recenly), her prce varaon s no parcularly large. 6 On he negave sde, here are a number of dffcules n esmang loan reurns. Frs, here are many ypes of bank loans, e.g., mos noably amorzng versus revolvng loans, wh varous feaures ncludng floang versus fxed paymens, bul-n prepaymen opons, rae reses based on a change of cred rsk of he borrowers, ec Second, gven hese ssues, we make he followng assumpons, namely ha erm loans amorze lnearly over her lfe, and, for revolvers, ha 0% s drawn down durng he year. For he sample perod whch he loan daa are concden wh he bond daa, reurns are calculaed usng loan prces and he neres over he monh. The hrd problem s ha, pror o November 999 or for a 4 For example, some problem areas are bonds beng eher called, convered, endered, repurchased wh snkng fund provsons, or exchanged n he case of Rule 44 secures, and so forh. 5 For he case where a bond prce s mssng for he monh, we nerpolae he bond prce assumng changes n relave proporon o oher bonds of he frm, he relave change beng deermned by s relave duraon. Inerpolaon occurs n 0.9% of he sample. 6 See, for example, lman (006) and charya, Bharah and Srnvasan (007) who documen very hgh recovery raes on bank loans and hus low losses gven defaul. 6

8 number of frms no covered n he prcng daase, we need o apply an alernave approach o generang loan reurns. Specfcally, snce boh he bonds and loans can be vewed as conngen clam s on he frm s asses, we run a panel regresson, broken down by frm rangs, of he excess reurn on a frm s bank loans agans excess reurns on he frm s bond porfolo and reasures (of smlar duraon o he bonds). These coeffcens are hen used o marx prce he loans of frms (and perods) whch bank loan daa are no avalable. 7 Thus, he reurn on a frm s asses s calculaed as R quy B Bond L Loan = R + R R, () sse + + B + L + + B + L B + L T T where s he marke value of equy, B s he marke value of he bonds, and L s he esmaed marke value of he loans. T + C. Sylzed Facs The mos mporan, and novel, daa n hs paper are he bond prce daa. Whle bank loans affec he leverage whn he frm, her dependency on changes n underlyng asse values s much less due her beng a he op of he capal srucure. Table provdes a summary of he qualy of he bond daa ha s used hroughou he sudy and was alluded o n Secon II.. above. The able breaks down each frm no dfferen rangs classes (from o CCC and below). For each class, we calculae he number of represened frms, he number of monhly bond observaons, he frequency by whch he bond prces do no change from monh o monh, he frequency by whch a leas one bond whn a frm does no change from monh o monh and hs frequency weghed by he amoun ousandng and he frm sze. Whle a zero bond prce change s suggesve of saleness, s by no means generally rue. For example, f expecaons of he probably of defaul and/or neres raes do no maerally change, hen one mgh expec a zero change. Neverheless, ha sad, across all bonds, hs ncdence occurs only 3.6% of he me. s he rangs decrease across frms, he probably ends o rse, reachng a peak of 4.6% wh B-raed frms. he frm level, s more lkely ha a leas one bond no change prce, e.g., 0.5% overall, wh hgh yeld frms of BB, B and CCC havng respecvely 7 The resuls are robus o varous specfcaons, mos probably due o he relavely low volaly of bank loans n he frs place. 7

9 7.86%, 5.63% and.58% ncdences. 8 When hese resuls are weghed by boh he amoun of bonds ousandng and he frm sze, hese ncdences drop dramacally o 6.33%, 5.7% and 7.4%, respecvely. Thus, can be correcly nferred ends o be an ssue wh much smaller frms. noher way o gauge he qualy of he daa s o look a he conemporaneous and leadlag auocorrelaon properes of he frm s bond, equy and asse reurns. For he enre sample and across each rangs class, Table B repors hese sascs. For example, he auocorrelaons of each frm s bond porfolo reurn are que small albe posve. Dependng on he number of bonds whn each frm, he posve number can be conssen wh some degree of nonradng as descrbed by Scholes and Wllams (978). Ineresngly, he frm s asse reurns frs pckup he auocorrelaon properes of he equy for he more hghly raed frms and move o hose of he bond reurns for he lower raed frms. Ths s que conssen wh he Black and Scholes (97) and Meron (974) vew of he frm s capal srucure. In fac, he conemporaneous correlaon beween equy and bond reurns of he frm s n he md eens percenage wse for hrough, and hen s 0.9, 0.39, 0.46 and 0.43 for BBB, BB, B and CCC, respecvely. Thus, he mplcaon ha deb looks more lke equy as he asses decrease n value (here represened by frm rang) holds rue. s a fnal check on he daa, Table B also repors varous lead-lag relaons beween equy and bonds. There s some evdence of a lead-lag relaon beween equy and bonds for he lower raed frms. Whle essenally zero up o BBB-raed frms, BB, B and C have 0.09, 0.0 and 0.5 correlaon a he frs lag respecvely. Whle hs could be slow response o nformaon across dfferen markes, could also represen some degree of saleness. One way o dfferenae saleness versus he marke segmenaon hypohess s o check wheher bonds also lead socks. The able shows smlar cross-correlaon paerns albe a lower magnudes, e.g., BB, B and C have 0.03, 0.04 and 0.05 a he frs lag respecvely. For eher lead-lag relaon, he correlaon drops o zero a he second lag. Coupled wh Table and B, s reasonable o conclude ha here exss a small, bu no major, degree of saleness a he monhly level. Gven he comfor level wh he daa, n hs paper, we look a he properes of wo dfferen seres: () qunle porfolos formed on leverage, and () ndvdual frms. 8 s an asde, he fac ha he occurrences are much hgher a he frm level suggess blanke marx prcng 8

10 . Porfolos Wh respec o he porfolo formaon mehod, consder he leverage porfolos. The sample perod covers March 99 o Ocober 007. For a frm o be ncluded n he porfolo, mus be a non-fnancal frm and have marke value of he asses o be a leas 50mm n December of he prevous year. In each December, frms are sored accordng o her leverage raos and hen held hroughou he year. We form qunle porfolos wh he frs porfolo beng frms wh zero leverage. fer each year, porfolos are reformulaed. Table 3 provdes summary sascs for he equy and asse reurns on hese wo ses of porfolo seres, n parcular, her mean, volaly and asse/equy rao. Consder frs he leverage porfolos. The zero leverage porfolo asde, he mean asse/equy rao over he sample perod s %, 35%, 68% and 85% respecvely for he levered porfolos. Monhly expeced reurns on equy ncrease smlarly from 0.46%, 0.63%, 0.65% and 0.8%, wh volaly a 3.8%, 3.7%, 3.7% and 4.5%, respecvely. 9 frs glance, one mgh be surprsed by he relavely fla paern of he volaly of equy reurns across levered porfolos. Ceers parbus, sandard heory would mply ha equy volaly should be ncreasng across leverage. Of course, he amoun of leverage s an endogenous choce by he managers of he frm. Faced wh a gven busness uncerany (.e., he frm s asse volaly), and f here are coss o fnancal dsress, hen one mgh expec he managers o choose leverage accordngly. s a frs pass, Table 3 shows ha hs s ndeed he case. cross he levered porfolos, monhly asse volaly drops from 3.4% o.8% o.3% o.7%. I may be ha leverage s opmally chosen o arge a specfc level of equy volaly, perhaps proxyng for a defaul probably. Whle hs deserves fuure research, hs resul s mporan because suggess one needs o be careful when nvesgang he rsk/reurn relaon n he crosssecon f porfolo sorng or addonal facors correlae o leverage.. Indvdual Frms probably does no occur. 9 The zero leverage porfolo s a lle anomalous here wh monhly expeced reurns of 0.9% and volaly of 7.8%. Ths perod ncludes he so-called nerne bubble and hus he zero levered frms have a sgnfcan echnology l. 9

11 Table 3B summarzes he mean and volaly of he equy and asse reurn of ndvdual frms by presenng her mean and medan n he overall sample and n he cross-secon of he 4 leverage porfolos. The resuls are smlar n spr o III.C. above. For example, across hese porfolos, he average frm equy volaly on a monhly level s 3.%,.3%,.% and 5.8%, respecvely. gan, whou seeng he porfolo resuls above, he fndng may be surprsng gven ha he average marke leverage rao for each frm s respecvely.9,.39,.75 and The above explanaon s ha leverage s a choce varable, and due o he mpac of asse volaly on he coss of fnancal dsress, he radeoff heory of capal srucure would sugges a negave relaon beween leverage and frm level volaly,.e.,.4%, 9.%, 7.5% and 6.3% as leverage ncreases. III. The Condonal Volaly of sse Reurns There s overwhelmng evdence ha he volaly of equy reurns s me varyng and perssen. Ths s rue a he marke ndex, porfolo and ndvdual frm level. Some of he earler leraure n suppor of hese fndngs nclude ngle (98), Bollerslev (986), Bollerslev, Chou and Kroner (99), and Bollerslev, ngle and Nelson (994), among many ohers. n addon, here s equally srong suppor for asymmery n he relaon beween volaly and reurn shocks. In parcular, volaly ncreases wh negave reurns. gan, hs resul s robus o ndex, porfolo and ndvdual frm daa (e.g., Nelson (99), Cheung and Ng (99), Glosen, Jagannahan and Runkle (993), and Braun, Nelson and Suner (995), among ohers). Whle researchers have employed varous models o capure hs asymmerc volaly relaon, he workhorse has ofen been he GRCH(,) model of Nelson (99) gven by R + log h = h + + ε + = ϖ + θε + γ ( ε ε ) + λ log h For our sample, Table 4 repors he esmaon resuls of an GRCH(,) for each of he fve levered porfolos. The four less levered porfolos show consderable perssence n volaly wh he GRCH coeffcen rangng from 0.90 o Ineresngly, he mos levered porfolo has a consderably smaller coeffcen, namely frs glance, hs resul s surprsng. Snce equy prces approxmaely follow a random () walk, one mgh expec ha he deb/equy rao, B +L T, s hghly perssen, herefore, 0

12 leadng o he mos levered porfolo havng he greaer perssence. However, leverage s a choce varable. I mgh be he case ha, as equy prces fall, frms acually delever, hus causng a qucker reverson n volaly. The mos mporan parameer for our purposes s he coeffcen on he asymmerc erm. Several observaons are n order. Frs, whle he coeffcen s negave for he four levered porfolos, s acually posve for he zero leverage porfolo. Ths suggess leverage plays an mporan role n he deermnaon of he well-documened asymmerc volaly resul. Second, hough negave, he sascal sgnfcance of he less levered porfolos s less han he usual levels for wo of he hree porfolos. Fnally, and mos crucally, he mos levered porfolo has a much larger coeffcen han he oher porfolos, boh n magnude (e.g., 66% hgher han he nex larges coeffcen) and n sascal sgnfcance (e.g., a -sasc of 3.57). The evdence above s conssen wh he sylzed fac ha sock reurn volaly rses afer sock prces fall. Our new fndng s he mporance of leverage whn he GRCH(,) analyss of sock reurn porfolos. Of course, here has been consderable debae and emprcal evdence generaed abou how much of he asymmerc volaly effec s due o fnancal leverage as a resul of he sock prce fall (.e., leverage effec ) versus me-varyng rsk prema (.e., volaly feedback ). mong he papers ha have analyzed hs queson are Black (976), Chrse (98), French, Schwer, and Sambaugh (987), Campbell and Henschel (99), Duffee (995), Bekaer and Wu (000) and Yu (005). Because we measure he acual reurns on he asses, we can addonal evdence o hs debae. Specfcally, Table 4B presens GRCH(,) esmaes for he reurns on he asses of he same fve levered porfolos. The unque aspec of hs analyss s ha he porfolos are n erms of he underlyng asses (.e., delevered), so, by consrucon, leverage canno be a facor. Frs, volaly perssence now appears smlar across he fve porfolos wh coeffcens rangng from 0.89 o Ths s conssen wh he deleveragng hypohess dscussed above for he mos levered porfolo. Second, and of parcular neres o he debae on leverage versus volaly feedback, he coeffcens on he asymmerc volaly par drop for every levered porfolo. For example, he coeffcen on he mos levered porfolo drops from o afer deleveragng. Thrd, whle he

13 asymmerc volaly coeffcens are no sascally sgnfcan, he coeffcens are of all four levered porfolos are sll negave, rangng from o Ths fnal fndng suggess ha, alhough leverage s a key facor n explanng he asymmery, here s some resdual asymmery remanng. s a way o breakdown he leverage and volaly feedback hypoheses more closely, we repea he GRCH(,) analyss of Table 4 a he ndvdual frm level.. Indvdual Frms For he ndvdual frm by frm GRCH(,) esmaon, due he amoun of nose n ndvdual equy and asse reurns, we requre he frm have () reasonable (.e, saonary) RCH and GRCH parameer esmaes 0, and () a leas 60 monhs of connuous daa. We also allow for non-gaussan error dsrbuons by ncludng -dsrbuons as a possbly due o he kuroc daa a he ndvdual frm level. Of he nal 7 frms, 853 reman ha sasfy all hese crera. Table 5 repors he mean and medan esmaes of he RCH, GRCH and asymmerc coeffcen for he GRCH(,) frm by frm esmaon n he overall sample, as well as across he fve groupngs based on leverage. The resuls are smlar n spr o he porfolo resuls provded n Table 4 n a number of ways. Frs, he mean GRCH parameer s around 0.9 across he varous groupngs for boh he equy and asse reurn esmaons. Second, he asymmerc coeffcens are negave across every porfolo groupng, confrmng he well-known resul. Thrd, for he less levered porfolos, here s no much dfference n he mean esmaes beween equy and asse reurns (e.g., for he leas levered porfolo, -0.4 versus -0.0, and for he nex, less levered porfolo, -0.7 versus -0.). In conras, for he more levered porfolos, he dfferences are magnfed (e.g., -0. versus for he second hghes levered porfolo, and -0.7 versus for he mos levered porfolo). s a fnal commen on hese ndvdual esmaes, o ge around droppng almos half he frms, we also perform a sacked regresson whch pus all he frm observaons ogeher. To adjus for dfferences n volaly levels, he volales are sandardzed across 0 Specfcally, we requre he GRCH coeffcens o be beween 0. and, and posve RCH coeffcens.

14 frms. Table 5 repors he se of parameers from hese GRCH(,) esmaes for boh equy and asse reurns. Conssen wh he mean and medan esmaes, he sacked esmaes show () a hgh level of perssence for boh he equy and asse reurns, () negave coeffcens on asymmery, and () a declne across he board n asymmery movng from equy o he asses, wh he greaes drops occurrng for he mos levered porfolos (e.g., o and -0.3 o , respecvely). Par of he movaon for lookng a ndvdual frms was o be able o separae he volaly feedback effec from he leverage hypohess. Specfcally, he volaly feedback sory argues ha, f marke volaly s prced and ncreases, hen he rsk premum wll also ncrease, leadng o a sock prce declne. Thus, he causaly beween ncreasng volaly and negave reurns s oppose o ha of he leverage effec. In order o separae he effecs, we run a one-facor model wh an dosyncrac GRCH(,) volaly: R R equy + asse + = β = βr m+ + h R m+ ε h ε + + where he marke value of he asses =B+L+, and Rm s he reurn on he unlevered marke from our sample (albe ncludng zero levered frms and fnancal frms). Noe ha equaon (3) removes he marke facor and herefore he volaly feedback effec as a possble explanaon. If volaly feedback were a prmary explanaon of he asymmery, hen he coeffcens on dosyncrac volaly should fall dramacally. Table 5B repors he resuls for he analyss of he dosyncrac volaly. The resuls are generally no good news for he volaly feedback effec. In parcular, he mean esmaes a he dosyncrac, frm level are only margnally lower han before, e.g., for equy, across he fve porfolos respecvely, from (-0.3, -0.3, -0.6, 0-. and -0.9) o (-0.0, -0.08, -0.3, -0., and -0.8). Smlar resuls hold a he asse reurn level. In general, here s a unform drop of around 0.03 across all he porfolos. These resuls are also confrmed a he GRCH(,) sacked esmaon of dosyncracc volaly for boh equy and asse reurns. On he posve sde, removng he marke porfolo causes a (3) The paral dervave n (3) s calculaed from he Black-Scholes formula and gven as ln ( K ) + ( r +.5 ) 3 = where / K s rao of asse value o face value of long-erm deb, r s -year reasury consan maury yeld, s asse volaly usng he full sample and T s face-value-weghed me-o-maury. T T

15 unform drop n all he asymmerc coeffcen esmaes. On he negave sde, he drop s que small. Coupled wh he prevous resuls a he porfolo level n Tables 4 and 4B, and wh he ndvdual resuls of Table 5, he fndngs here n Table 5B sugges he facors n order of mporance for explanng asymmerc volaly s leverage, hen an unspecfed resdual, and fnally he volaly feedback (.e., me-varyng rsk premum). IV. The Condonal Volaly of sse Reurns: Srucural pproach The evdence gven above suggess ha leverage s an mporan componen for explanng me-varaon n volaly, especally wh respec o he sylzed fac ha volaly ncreases when he underlyng sock prce falls. The unresolved queson from he analyss n Secon III s jus how mporan s leverage? Suppose he assumpons underlyng Black and Scholes (97) and Meron (974) hold so ha asse reurns follow a geomerc Brownan moon, neres raes are consan, here are no mpedmens o arbrage, and ha he frm s capal srucure can be collapsed no equy plus one ssue of zero coupon deb (wh a maury ha maches he duraon of he acual daa). Then, for any frm, we can sar wh he basc resul from Black-Scholes- Meron ha R = R. ggregang o he porfolo level, s possble o show ha w R = w R, (4) where w represens he marke value weghs of he equy and asse porfolo, respecvely. To see hs, noe ha 4

16 5 Usng equaon (4), we can hen model he condonal volaly or log condonal volaly of porfolo reurns on equy n erms of wo facors: () he porfolo s leverage, or specfcally oal asses o equy, and () he condonal volaly of he porfolo s adjused asse reurns: ( ) ( ) = + + ~ : : R w vol R w vol, (5) where he adjused asse reurn R R = ~. We can look a changes such as ( ) ( ) ~ log log log + = (6) o beer undersand he volaly properes of equy porfolos. s a frs pass, Fgure graphs sde by sde he GRCH(,) esmaed volaly of he four levered porfolos and her marke rao of asses/equy over he enre sample perod. Several feaures of hese graphs capure he more dealed analyss o follow. Frs, gven ha he asse volaly and leverage rao ener no equaon (5) n equal proporon, he graphs show mmedaely ha asse volaly s he more mporan facor. cross all he porfolos, even he mos levered one, asse volaly vares by a mulple more han leverage. Whle hs s parly due o consrucon (.e., he porfolos are rebalanced yearly n erms of leverage), neverheless shows ha equy volaly s me-varyng properes are for he mos par due o he underlyng asses. (Noe ha he rebalancng ssue s addressed n he subsecon below when we look a ndvdual frms.) Second, leverage raos are much ( ) ( ) ( ) = = = = = R R R R R R R R R R.

17 more perssen han asse volaly hrough me. Ths mples ha, even hough asse volaly s he predomnan source of me-varyng equy volaly, leverage has long-erm effecs. Thus, a shock n asse values ha ncreases boh he leverage and he underlyng asse volaly wll have long- and shor-erm mpac, respecvely. Fnally, alhough hese resuls hold across he four levered porfolos, s clear ha he mos levered porfolo has more neresng properes, such as s leverage rao s more varable and asse volaly appears less perssen. In hree separae panels, Table 6 presens summary sascs for he four levered porfolos n erms of he breakdown beween leverage volaly and adjused asse reurn volaly. Usng equaons (5) and (6), Panel drecly compares he varably of asse volaly and leverage n boh levels and changes across he four porfolos. Panel B presens he auocorrelaon properes of asse volaly, leverage and equy volaly mpled by he srucural model a monhly lags -3, 6 and. Usng he srucural models n equaons (5) and (6), Panel C drecly calculaes he proporon of me-varyng equy volaly ha can be explaned by asse volaly and leverage, respecvely. Wh respec o Panel, boh n levels and dfferences, he volaly of asse volaly s much greaer han he volaly of marke leverage raos. Of course, par of hs explanaon may be due o measuremen error n our asse volaly esmaes. Neverheless, n levels, he volaly of asse volaly versus fnancal leverage s (0.7, 0.5, 0.9 and 0.4) versus (0.03, 0.05, 0.09 and 0.6) respecvely across he four leverage porfolos. From he srucural pon of vew, n erms of he ably o explan he me-varyng properes of equy volaly, asse volaly s herefore necessarly he more mporan facor. Panel B shows ha, n general, leverage s more perssen han asse volaly. a frs look, he mos levered porfolo asde, he frs order auocorrelaon suggess smlar properes, e.g, for he four levered porfolos respecvely, he auocorrelaons are (0.95, 0.97, 0.97 and 0.96) for her leverage componen and (0.93, 0.88, 0.97, and 0.5) for her asse volales. When he auocorrelaons, however, are exended o 6 and lags respecvely, he resuls look que dfferen. For example, a he h lag, he auocorrelaons are (0.8, 0.75, 0.74 and 0.63) for her leverage componen whle only 6

18 (0.57, 0.38, 0.68, and 0.05) for her asse volales. Ths necessarly means ha shocks o asse prces affec volaly boh n he shor- and long-erm albe hrough dfferen mechansms, namely he ransory properes of asse volaly and more permanen shocks of fnancal leverage. Ths s que noceable when we use he srucural model of equaon (5) o esmae he auocorrelaon of equy volaly. The auocorrelaon wll be a funcon of he ndvdual auocovarances as well as he varances of he leverage componen and asse volaly. Snce he varance of asse volaly s much hgher, he nal auocorrelaon properes of equy volaly ake on he underlyng asse volaly, only o evenually ake on prmarly he properes of leverage. For example, consder he mos levered porfolo. The auocorrelaons of equy volaly mpled by he srucural model over lags,, 3, 6 and are respecvely 0.68, 0.47, 0.39, 0.40, and 0.8. smlar paern holds across he oher porfolos. Ths resul may help explan he well-known sylzed fac ha volaly has boh a mean-reverng sandard GRCH-lke represenaon wh a long memory componen (e.g., see Bollerslev and Mkkelsen (996), ngle and Lee (999), dran and Rosenberg (008) and ngle and Rangel (008)). Ths long-erm dependence asde, Panel C drecly calculaes he proporon of condonal equy volaly explaned by he leverage componen versus he underlyng volaly of he asses. These calculaons are performed n boh levels and changes n volaly usng equaons (5) and (6). In levels, perhaps no surprsngly, he relave mporance of leverage for explanng equy volaly ncreases wh leverage. For example, from he low o hgh levered porfolos, he conrbuon goes from -0.4% o 0.0% o 3.4% and o 34.7%, respecvely. We noed above ha, by rebalancng he porfolo every year no one of fve qunle porfolos, we mgh be removng some neresng dynamcs of leverage a he ndvdual frm level. Moreover, would be nce o be able o furher breakdown he relave proporon of explaned equy volaly. For example, along wh he aforemenoned leverage versus me-varyng rsk prema debae, how much of me-varyng equy volaly s explaned by marke versus dosyncrac movemens?. Indvdual Frms: Srucural smaes 7

19 The dervaons of he srucural model a he ndvdual frm level follows smlarly o ha a he porfolo level descrbed a he begnnng of hs secon. ssumng he Black- Scholes-Meron ype assumpons, s possble o show ha where N( d ) ( K) ( r ) + ( d) = N, (7) ln +.5 T =, T s he maury of he deb, r s he rskless rae and K T s he face value of he zero coupon deb. Followng along he lnes of he above analyss, we can eher model he volaly or log volaly usng he above me seres mehods, and evaluae he properes of he prcng errors, such as unbasedness, mean-squared error (.e., r-squareds). We can also look a changes: ( ) = log N( d ) + log( ) log. (8) Table 7 presens summary sascs for ndvdual frms n he overall sample and across he dfferen leverage groupngs. s n Table 6, we focus on hree panels coverng he relave varaon of he leverage and asse volaly componens, he perssence properes of hese componens, and her esmaed conrbuon o equy volaly. From Panel, he varaon of he frm s leverage componen versus s asse volaly s que large relave o he aforemenoned resuls for porfolos. Ths s rue across all leverage groupngs alhough clearly s mos prevalen for he hgher levered frms. In levels, average volaly of fnancal leverage versus asse volaly for ndvdual frms s (0.08, 0.4, 0.0, 0.39) versus (0.6, 0.6, 0.6, 0.8) respecvely across he four leverage groups, whereas s porfolo level counerpar s (0.03, 0.05, 0.09, 0.6) versus (0.7, 0.5, 0.9, 0.4). The basc premse here s ha, n erms of he srucural model, he daa suggess ha leverage plays an mporan role n deermnng he me-varaon of equy volaly. The mos lkely explanaon for he conras wh he porfolo resuls s ha he rebalancng of he porfolos reduces he effec of whn-frm changes n leverage as hese frms move from one levered porfolo o he nex. Table 7, Panel B shows ha, smlar o he porfolo resuls, he perssence of he leverage pece s much greaer han ha of he frm s asse volaly. For example, a he 8

20 h lag, he mean auocorrelaons for leverage are (0.4, 0.39, 0.40, 0.40) whereas her asse volaly counerpars are (0.37, 0.3, 0.35, 0.4). Thus, equy volaly a he frm level has wo componens, a ransory one drven by he varaon n he underlyng asses, and a more permanen one drven by fnancal leverage. Gven he well-documened asymmery n volaly, he mos lkely source for boh hese componens s he same facor, namely negave shocks o he underlyng asses. Panel C presens esmaes of he relave conrbuon of he leverage componen and asse volaly o varaon n he frm s equy volaly. The resuls are presened for he mean and medan esmaes for boh levels and changes n volaly, usng eher he ndvdually esmaed GRCH(,) coeffcens or he sacked esmaon. 3 s he leverage of he groupng ncreases, he conrbuon of leverage owards he me-varaon n equy volaly also ncreases, e.g., n levels, from.8% o.% o 3.% o 44.4%, and, n changes, from.% o.8% o 34.% o 55.8%. Ths basc fndng s robus o wheher we use medans or he sacked GRCH(,).. Impled quy Volaly and Is Deermnans In he analyss so far, we have looked a equy volaly mpled by he model srucure gven by equaons (7) and (8). Ideally, would be nce o relax he srucure and relae how much of he rue me-varyng equy volaly could be explaned by asse volaly (and s ndvdual componens) and fnancal leverage. Ths s mporan because a number of assumpons wen no he dervaons of (7) and (8). The problem, of course, s ha our esmaes of me-varyng equy and asse volaly use some of he same underlyng daa (e.g., equy reurns) and he GRCH framework. Thus, regressng esmaes of asse volaly on equy volaly wll nvolve consderable common measuremen error. s a way around hs problem, we colleced -monh mpled volales from a-he-money opons on he equy for as many of he frms n our sample as possble from Oponmercs. Ths reduces our sample sze from 647 frms (90 f sacked) o 554 frms (3 f sacked); n oher words, he concden sample of mpled equy volales and monhly asse volaly esmaes s abou 86% as large. The nce feaure of hs approach s 3 Recall he movaon for usng he sacked GRCH was ha allowed us o use he full sample of frms, whle abou one-half he sample of ndvdual esmaons had o be dropped due o nonsaonary esmaes of GRCH. 9

21 ha he daa sources are que dfferen, namely opons daa on he frm versus he reurn on he frm s asses (derved from equy, bond and loan daa). 4 s a frs look a he daa, we run he followng regresson: log ( ) = φ log N( d ) + γ log( ) ( ) = θ log N( d) + λ log( ) + + ε, (9) log η where equy volaly,, s he monhly mpled volaly from opon markes, asse volaly,, s esmaed from an GRCH(,) usng asse reurns, and / s he marke leverage of he frm. The nal resuls are repored n Table 8 for boh he mean and medans of he regressons, as well as he sacked regresson where we esmae one se of coeffcens. Whle he srucural heory of equaons (7) and (8) mply coeffcens of and R-squareds of 00% f here were no measuremen error, Table 8 provdes mpressve resuls noneheless. The regresson esmaes of (9) are based downward, hoverng around 0.5 for leverage and 0.7 for asse volaly wh R-squareds of approxmaely 55%. These resuls are robus across he dfferen leverage cross-secons. Moreover, f we use he sacked regresson esmaes, he coeffcens are much closer o he heorecal value of, especally for asse volaly. Ths s conssen wh measuremen error n he GRCH esmaon of asse volaly. Perhaps, no surprsngly, he resuls of he dfference regressons are weaker hough smlar n spr. The R-squareds drop precpously o beween 0%-5% wh a correspondng fall n he coeffcen of asse volaly. Wha do hese resuls mean n erms of wha drves me-varyng equy volaly? Table 8B repors he varance decomposon of equy volaly n erms of s explaned poron. Smlar o prevous resuls, asse volaly s he domnan facor for frms wh low leverage. Ths s rue for boh levels and dfferences of volaly, and wheher we measure he mean or medan whn he sample, or run a sacked regresson. For example, s mean percenage conrbuon for equy volaly levels s on average 8%, 70%, 69% and 57%, respecvely as leverage ncreases. Moreover, for he sacked regresson, he resuls are 4 For he GRCH parameer esmaes of asse volaly a any gven pon n me, we use he enre sample excep for he perod mmedaely surroundng he mpled volaly. 0

22 smlar albe weaker, 70%, 67%, 55% and 50%, respecvely. Fnancal leverage s herefore sll an mporan deermnan, especally when frms have hgh leverage. quaon (3) of hs paper separaed asse volaly no wo componens, namely marke-wde and dosyncrac asse volaly. We can rewre hs equaon n erms of equy volaly by leverng up usng he adjused leverage rao, ha s,, = L, β M, + L, (0) where he β s he asse bea of frm, M s he GRCH esmaed varance of he asse reurn on he marke, s he GRCH esmae of he dosyncrac volaly of he asses, and L = s he adjused leverage rao of asse. The above equaon (0) can be log-lnearzed and rewren n he followng erms: log(, ) e log( L ) + log( x M ) + log( x, ) + e + e x + k () where we have suppressed he frm subscrps, k s a consan, and x s assumed o be saonary and gven by he formula x = log. 5 β M There are several mplcaons of equaon (). The coeffcen on he log of adjused fnancal leverage s one, whle he oher wo coeffcens on he asse volaly of he marke and dosyncrac asse volaly sum up o one (and herefore boh coeffcens are less han one). Of course, he quany x dffers across frms, so hese coeffcens wll vary across frms as well, he bea of he asses beng an mporan deermnan of hs. 5 To see hs, noe ha by akng logs of equaon (0), we ge log(, ( ) + + β log ) = log( L ) + log M β M. Now expand he las erm around he mean of = x x log log( β, ) = log( L ) + log β + log M + log + e can be M wren as approxmaely x x e log(, ) log( L ) + log( β ) + log( M ) + log( + e ) + x ( x x) ) e. Subsung n for x, + rearrangng erms, dvdng boh sdes by, and collapsng he non me-varyng erms no he consan k, we ge he desred resul n equaon ().. Thus, ( ) ( ) ( )

23 The op rows of Table 9 provdes he resuls for he regresson of me-varyng asse volaly of he marke, dosyncrac volaly of he frm s asses, and he frm s marke leverage on he mpled volaly of he frm s equy. In levels, he varables capure que well he varaon n mpled equy volaly. The R-squareds across he four leverage groupngs are all n he 60+% range. The average coeffcens on marke and dosyncrac asse volaly do no que sum o, bu are n he range of 0.79 o Ineresngly, he coeffcens are of smlar magnude. s wh prevous ables, leverage also plays an mporan role, albe n he 0.37 o 0.48 range, somewha far from s heorecal value of. The op panel of Table 9B provdes he decomposon resuls. For he low leverage groupngs, leverage has only a small mpac, e.g., 8%, bu grows seadly wh leverage, from 8% o 34%. 6 The remanng componens are somewha spl beween marke and dosyncrac asse volaly. The resuls n mpled volaly dfferences are also shown n Tables 9 and 9B. The R-squareds are n he 5% range. The coeffcens are smlar n magnude o he resuls n levels, wh perhaps a srenghenng of he effec of changes n leverage and a weakenng effec of he changes n marke and dosyncrac volaly. s menoned n secon III, here s consderable neres n ryng o beer undersand he asymmerc volaly relaon, and, n parcular, he mporance of leverage versus rsk prema versus dosyncrac (possble behavoral ) effecs. Currenly, our approach and ohers s o es for how bg hese effecs are whn a GRCH-lke framework. n alernave approach would be o see wheher hese esmaed effecs acually explan he mpled volaly levels or changes hrough me. Noe ha equaon () can be rewren n erms of he ndvdual componens of he GRCH(,) esmaes of marke and dosyncrac asse volaly. Tha s, log log log Mk mk mk log do g do h do = a l + b + cε + d ε + f + ε + ε () where he coeffcens a-h are no mposed by he model n () and are allowed o be unconsraned. We can also calculae equaon () n changes, specfcally akng he dfference, 6 The sacked regresson resuls suppor an even sronger leverage componen o a frm s mpled equy volaly. Moreover, across he groupngs from low o hgh leverage, he sum of he volaly coeffcens are que close o, equalng respecvely 0.98, 0.90, 0.93 and 0.9. These resuls are presened n Tables 9C and 9D.

24 equy Mk Mk do log = a logl + b(log log ) + c(log log Mk ), and pluggng n he GRCH(,) esmaes for marke and dosyncrac asse volaly, we oban he followng : log log log Mk mk mk log do g do h do = a l + b + cε + d ε + f + ε + ε. (3) The erms n equaons () and (3) have a clear nerpreaon n erms of he mpac of he heores underlyng asymmerc volaly. ε Mk 3 do and ε represen he shock o he marke reurn of he asses and o he dosyncrac reurn of he frm s asses. The former represens he me-varyng rsk prema effec commonly ermed he feedback effec, whle he laer s a more puzzlng dosyncrac componen ha some researchers mgh denoe behavoral. In erms of oher major componens, l represens he leverage effec, and he lagged Mk volales, and do, are he perssen effec of marke and dosyncrac asse volaly. For example, n equaon (3), he lagged volaly erms represen he decay effec gven ha volaly s n fac mean reverng. The boom rows of Table 9-9D provde he resuls for he regressons n equaon () and (3). The level regressons all produce sgns and, o some exen, coeffcens n he drecon of our nuon. Fnancal leverage, he mos recen marke volaly and dosyncrac volaly all come n posve and large n magnude. These are he major effecs n erms of explanng me-varyng equy volaly. s an llusraon, consder he mos levered groupng of frms; he varance decomposon shows ha leverage, lagged marke asse volaly and lagged dosyncrac asse volaly explan 34%, 9% and 9%, respecvely of he explaned varaon wh an R-squared of 68%. Ths s no o mply ha he lagged reurn (asymmerc shock) or lagged absolue reurn (volaly shock) are no mporan. In fac, her sgns,.e., a negave asymmerc shock and posve volaly shock, across all groupngs, mean versus medan, and sacked regressons are generally conssen wh her hypohess. However, he resuls clearly show ha hey are no crucal for undersandng he level of volaly. These resuls, however, are compleely reversed for changes n equy volaly. Though mean reverson n curren volaly has he poenal o be an mporan deermnan of volaly changes, he effec s close o zero. ll he varaon s now due o he

25 asymmerc shocks,.e., lagged reurn componens, or o changes n fnancal leverage. For example, for he mos levered groupng (whch s ypcal of he oher groupngs as well), he relave conrbuon o equy volaly changes s 6%, 7%, 0%, 39% and % respecvely for changes n fnancal leverage, lagged marke reurns, lagged marke absolue reurns, lagged dosyncrac reurns, and lagged absolue dosyncrac reurns. Thus, dosyncrac shocks o asse reurns have a large mpac on condonal volaly changes. Ths resul presens a sylzed fac ha needs o be explaned. Gven ha hese shocks are, for he mos par, dversfable, s no clear why he condonal volaly of equy responds. Ths pon asde, a common feaure of boh he level and change regressons for mpled volaly, however, s he connued mporance of fnancal leverage. V. Concluson Usng a unque daase of equy, bond and loan reurns a he frm level, we are able o measure a frm s asse reurns and esmae he volaly of a frm s asses. Ths allows us o more drecly nvesgae he mpac of fnancal leverage on he equy volaly of he frm. n overall concluson from hs sudy s ha fnancal leverage s mporan for explanng movemens n equy volaly. Ths s rue a he ndvdual frm and porfolo levels, and s robus o numerous specfcaons. The resuls from hs paper also show, however, ha asse volaly self me-vares and, excep for he mos levered frms, s he domnan facor. Some of he resuls n hs paper sugges valuable areas of fuure research. Frs, he me-varaon of boh fnancal leverage and asse volaly argues for perhaps a more fundamenal approach o analyzng asse prcng heores relang equy reurns o marke facors. Tha s, he leraure should ake a more serous look a unlevered reurns. Second, anoher fndng, namely ha leverage has more a permanen mpac on equy volaly han he ransory (albe large) effec of asse volaly, mples neresng dynamcs a shor and long horzons ha should furher be explored. Thrd, we documen mporan dosyncrac effecs a he asse level on equy volaly levels and changes. reasonable queson s wha ype of model can produce hese effecs. Lasly, he sylzed fac ha leverage s nversely relaed o asse volaly has mporan mplcaons for corporae 4