A New Method to Measure the Performance of Leveraged Exchange-Traded Funds

Transcription

1 A ew Mehod o Measure he Performance of Leveraged Exchange-Traded Funds Ths verson: Sepember 03 ara Charupa DeGrooe School of Busness McMaser Unversy 80 Man Sree Wes Hamlon, Onaro L8S 4M4 Canada Tel: (905) Ex Fax: (905) E-mal: charupa@mcmaser.ca and Peer Mu DeGrooe School of Busness McMaser Unversy 80 Man Sree Wes Hamlon, Onaro L8S 4M4 Canada Tel: (905) Ex. 398 Fax: (905) E-mal: mupee@mcmaser.ca

2 A ew Mehod o Measure he Performance of Leveraged Exchange-Traded Funds Ths verson: Sepember 03 Absrac We examne he effecs of daly reurn compoundng, fnancng coss, and managemen facors on he performance of leveraged exchange-raded funds (LETFs) over varous holdng perods. We also propose a new mehod o measure LETFs' racng errors ha allows us o dsenangle hese effecs. Our resuls show ha he compoundng effec generally has much more nfluence on racng errors han oher facors do, especally for long holdng perods and n a "sdeways" mare. Besdes, he explc coss (.e., he expense raos) and oher facors (e.g., fnancng coss) can maerally affec he performance of LETFs, especally for hose wh hgh leverage raos and bear funds. JEL classfcaon: G0; G; G3 Keywords: Exchange-raded funds; Leverage; Tracng errors; Regresson analyss; Bull funds; Bear funds

3 . Inroducon A leveraged exchange-raded fund (LETF) s a publcly-raded fund ha promses o provde daly reurns ha are n a mulple (posve or negave) of he reurns on an ndex. To mee ha promse, he fund uses leverage, whch s ypcally obaned hrough dervaves such as fuures conracs, forward conracs and oal-reurn swaps. The amoun of leverage has o be adjused daly so ha he proporonal leverage (and hus he mulple) remans consan from day o day. In he US mare, he frs LETF was nroduced n 006. Snce hen, hey have successfully araced nvesor neres. As of he end of 0, here were over 00 LETFs wh oal asses of approxmaely $30 bllon. Ther underlyng ndces nclude soc ndces, bonds, currences, commodes and real esae. The avalable mulples are double leverage (+x and - x) and rple leverage (+3x and -3x). LETFs are no for long-erm, buy-and-hold nvesors. Ths s because he consan mananng of her leverage raos wll cause her long-erm compounded reurn o devae from he mulple of he underlyng ndex reurn over he same perod. The magnude and he drecon of he devaon depend on he lengh of he holdng perod and he pah ha he underlyng benchmar aes durng ha perod. For a gven holdng perod, he hgher he mulple (n absolue erms) or he more volale he underlyng ndex reurns (or boh), he greaer he chance ha a LETF's realzed reurn wll dffer from s saed mulple. The performance of exchange-raded funds s measured by her racng errors, whch are commonly defned as he devaons of he reurns on her ne asse values (AVs) from he reurns on her underlyng ndces. The smaller her racng errors, he more successful he funds are n machng he reurns on he ndces. Convenonally, here are a few ways o esmae a fund's racng errors such as by: () calculang he sandard devaon of he dfference beween he fund's AV reurns and he reurns on s underlyng ndex; and () regressng he fund's AV reurns on he underlyng ndex's reurns. These approaches are commonly used n sudes on radonal (.e., +x) ETFs and ndex muual funds (e.g., Frno and Gallagher, 00; Elon, Gruber, Comer and L, 00; and Gasneau, 004). However, hey can lead o ambguous resuls when used wh LETFs and nverse ETFs (hereafer, LETFs and In addon o hese LETFs, here are also nverse ETFs, whch provde daly reurns equal o he negave of he underlyng ndex reurns (.e., mulple = ). 3

4 nverse ETFs are collecvely referred o as "LETFs"). Ths s because he racng errors of hese funds are dcaed no only by facors ha are under he conrol of he fund ssuers (e.g., fees, expenses, replcaon echnques, ransacon coss, and he accrual of cash), bu also by facors ha are ousde of her conrol (e.g., he compoundng effec and fnancng coss). Accordngly, when evaluang he performance of LETFs, s mporan o undersand how he compoundng effec, fnancng coss, and he managemen facors nfluence he funds' racng errors, and o be able o separae he dfferen sources of errors. In hs sudy, we analyze hese effecs and propose a new mehod o measure racng errors ha explcly conrols for he compoundng effec. We hen apply hs mehod o a sample of nverse, doubleand rple-leveraged funds. We show ha, dependng on he reurn pahs of he underlyng ndces, he effec of compoundng can consue a very large poron of he oal racng errors. By screenng ou he compoundng effec, our proposed mehod s more precse and approprae han he convenonal measures for examnng he performance of LETFs. We also consder an exenson of our proposed mehod ha allows us o smulaneously conrol for any dfference n he fnancng coss nvolved n delverng he arge leverage reurns. By enablng us o solae he poron of he racng errors ha can be fully arbued o managemen facors, our proposed mehodology provdes us wh he necessary mercs o compare he managemen effcency of dfferen LETFs. As LETFs are relavely new producs, he leraure on hem s sll hn bu s qucly expandng. Some sudes examne he reurn dynamcs of LETFs (e.g., Avellaneda and Zhang, 009; Carver, 009; Cheng and Madhavan, 009; Gese, 00; Jarrow, 00; Lle, 00). Ohers nvesgae he prcng effcency of LETFs (.e., premums/dscouns relave o her ne asse values) and her racng errors (e.g., Lu, Wang and Zhang, 009; Guedj, L and McCann, 00; Charupa and Mu, 0, 03; Lovsce, Tang and Xu, 0; Shum and Kang, 03). Our paper adds o he laer group of leraure. The paper s organzed as follows. In he nex secon, we explan how he racng ably of LETFs can be affeced by varous managemen facors, fnancng coss, and he compoundng of daly reurns. Secon 3 descrbes our sample, whle Secon 4 repors he resuls of our prelmnary nvesgaon no he emprcal effec of compoundng on LETFs' racng errors. We presen our proposed measures of racng errors n Secon 5, and apply o our sample. Fnally, Secon 6 concludes he paper. 4

5 . LETFs' racng ably The ably of an LETF o rac s underlyng ndex depends on a number of facors. The frs group of facors s drecly relaed o how he fund s managed, and hus s generally under he conrol of he fund's managemen. They nclude nvesmen advsory and managemen servce fees, ransacon coss, managemen of dvdend dsrbuon, and he choce of replcaon sraegy. In hs paper, we refer hem as managemen effecs. We expec hose funds ha are more effcenly managed wll have smaller racng errors as a resul of hese managemen effecs. In order o have an accurae assessmen of a fund's managemen effcency, we need o conrol for oher facors ha may also nfluence he fund's racng ably, bu are no under he full conrol and dscreons of he fund's managemen. These oher facors nclude he fnancng effec and he compoundng effec. The former s he resul of he fnancng requremen of he funds (or her swap counerpares) n delverng he leveraged or nverse reurns. I s herefore governed by he fund's leverage mulple and he prevalng mare neres rae. The laer s he drec resul of he daly rebalancng of he fund's exposure n order o manan s arge daly leverage mulple. I s dcaed by he reurn pah realzed by s underlyng benchmar ndex. In he res of hs secon, we descrbe hese hree ypes of effecs n deal and how a fund's racng ably can be affeced by each of hem. In Secon 5, we propose a mehodology ha allows us o screen ou he fnancng and compoundng effecs n measurng he racng errors of LETFs. By dong so, we can more accuraely compare he managemen effcency of dfferen LETFs...Managemen effecs Invesmen advsory and managemen servce fees Smlar o radonal (.e., +x) ETFs, nvesors of LETFs bear he explc coss of managng he funds. The larges componen of hese coss s he nvesmen advsory and 5

6 managemen servce fees charged by fund ssuers. They are ypcally charged a annualzed raes based on he fund's daly ne asse values. For example, ProShares Ulra S&P 500, he larges +x LETF racng he S&P 500 ndex, has an expense rao of 0.9% per annum n 0. The hgher he expense rao, he more an LETF s expeced o underperform s underlyng ndex. The expense rao, however, does no capure all he coss ncurred n managng a fund. We dscuss hree of hese oher (mplc) coss below. Transacon coss The ransacon coss ncurred by he fund n generang he arge reurns are no refleced n he fund's expense rao. 3 These coss are ransacon fees for enerng no and modfyng dervave conracs. They are generally hgher for LETFs ha () rac ndces ha are more volale or less lqud; () have hgher leverage raos; and () have more frequen creaon/redempon of her uns. Furher, for LETFs ha, n addon o enerng no dervave conracs, also hold he underlyng secures of he ndces, here are ransacon coss for buyng and sellng he underlyng secures. 4 I s no uncommon for bull LETFs racng domesc equy ndces o have a poron of her exposure nvesed n he consuen socs of he ndces. 5,6 These LETFs ncur ransacon coss assocaed wh machng he changes n he ndex weghs and composons. Ths ype of ransacon coss has been shown o be one of he facors ha explan he racng errors of radonal (.e., +x) equy-based ETFs and equy ndex funds n general (e.g., Elon, Gruber, Comer and L, 00; Gasneau, 00; Frno and Gallagher, 00; Frno, Gallagher, euber and Oeomo, 004). Dvdends Oher explc coss may nclude admnsraon fees, cusodan fees, lcensng fees, and rusee fees. These fees neverheless represen a relavely small fracon of he oal explc coss. 3 Ths s no he ransacon coss (e.g., commssons and broerage fees) pad by nvesors of LETFs n buyng and sellng uns of LETFs on he exchanges. 4 For equy LETFs, here are poenal benefs for havng par of he exposure n he form of a porfolo n he consuen socs. For example, he fund ssuer can generae ncome from lendng he socs n he porfolo. 5 For example, n 0, ProShares Ulra S&P 500 generaed s 00% exposure o he S&P 500 ndex by a combnaon of equy secures mmcng he composon of he ndex (73%) and swaps and fuures conracs (7%). (May 0). 6 Unle bull funds, mos bear LETFs on equy ndces use only dervave conracs o acheve he desred exposures. 6

7 Sudes on radonal ETFs and ndex funds (e.g., Elon, Gruber, Comer and L, 0; Frno, Gallagher, euber and Oeomo, 004) show ha racng errors can occur f dvdends from he consuen socs are accrued n a cash accoun before hey are dsrbued by he fund provders o he un holders, resulng n an opporuny cos (for renvesmen). A smlar cos can be expeced o occur wh bull LETFs ha nves a poron of her exposures n he consuen socs of he benchmar ndces. Ths mplc cos s an ncreasng funcon of he dvdend yeld of he ndex, he amoun of me delay of dsrbuon, he conemporary reurn on he underlyng ndex, and he leverage rao. The effec, however, s unlely o be economcally sgnfcan. Ths s because only a poron of he exposure s generaed by holdng he consuen socs. More mporanly, he dsrbuable amouns of dvdends from he funds are usually much less han he acual dvdends receved. Typcally, any dvdends or neress receved by he funds are frs used o offse he advsory and managemen servce fees (on a daly bass) before hey are accrued and dsrbued o he un holders (see ProShares, 0). Replcaon sraeges The fund's choce of replcaon echnque can also play a role n affecng s racng errors. Whle he exensve use of dervave conracs (e.g., fuures, forward, or swaps conracs) s expeced o enable fund ssuers o replcae he leverage reurns reasonably accuraely, here are rss assocaed wh her use (e.g., bass rss and correlaon rss) ha can cause msracng. 7 In addon, he use of a porfolo of dervaves (o dversfy counerpary cred rss) may furher ncrease he chance of devaon from he nvesmen objecve. For LETFs ha nves par of her exposure n he underlyng secures, racng errors can occur f he funds do no use full replcaon, and only nves n a subse of he consuen socs based on some opmzaon sraeges...fnancng effec 7 In commenng on such rss, was saed n noe 3 of ProShares (0) ha: "When a Fund uses dervaves, here may be mperfec correlaon beween he value of he reference asse(s) and he dervave, whch may preven he Fund from achevng s nvesmen objecve." 7

8 To generae leveraged reurns, LETFs use leverage embedded n dervave conracs or (rarely) acual fnancal leverage. The performance of LETFs s herefore affeced by he fnancng coss assocaed wh he leverage. For bull LETFs (.e., funds wh posve leverage mulples), hey have o acqure fnancng o leverage up her long posons. For example, o creae wo mes he ndex reurn, a +x LETF s effecvely borrowng one me s exposure (va he counerpary of a dervave conrac); whereas a +3x LETF s effecvely borrowng wo mes s exposure o fulfll s leverage objecve and hus ncurrng wce he fnancng coss as he +x LETF. Therefore, he hgher he fnancng rae and he hgher her leverage raos, he more negavely her performance wll be affeced by he fnancng effec. 8 On he oher hand, bear LETFs (.e., funds wh negave leverage mulples) hold shor posons and so hey benef from hs fnancng effec. 9 The hgher he fnancng rae and he larger her leveraged raos, he more posvely her performance wll be affeced by he fnancng effec. More dealed descrpons of he fnancng effec can be found n Secon Compoundng effec In addon o he facors menoned above, racng errors can be caused by he effec of compoundng of daly leveraged reurns. Because he goal of an LETF s o generae he saed mulple of he reurn of he underlyng ndex on a daly bass, he fund has o rebalance daly s exposure o he ndex so ha he leverage rao remans consan. The compounded reurn over a holdng perod longer han one day can devae from he underlyng ndex reurn mulplyng by he promsed leverage even f here s no racng error on a daly bass. The magnude and he drecon of he devaon depend on he lengh of he holdng perod, he leverage rao and he reurn pah (.e., level and volaly) of he underlyng benchmar durng ha perod. To have an dea abou he effec of compoundng on LETFs' racng ably, we presen n Table a smple -day example of a +x and a +3x LETFs under fve reurn pahs (a formal dscusson follows shorly). For llusrave purposes, we assume here s no racng error on a daly bass, so we can focus our aenon on he effec due o compoundng. 0 8 The fnancng rae s commonly benchmared agans he LIBOR rae. 9 Ths mples ha he counerpares of he dervaves ransacons pass along he benefs from he shor posons o he funds. Ths fac s confrmed by fund ssuers such as ProShares (see Proshares, 0). 0 Tha s, we gnore he fnancng and managemen effecs as menoned n he prevous subsecons. 8

9 Inser Table Here Consder frs he +x LETF n Panel A. Under reurn pah #, he underlyng ndex ncreases by 0% on day, and declnes by 5% he nex day. Over wo days, he ndex reurn s (. x 0.95) = 4.50%. Ths may cause some holders of he +x LETF o expec ha hey would ge wce he ndex reurn or 9%, whch s wha an nvesor would ge f he/she used margns (.e., wh no daly rebalancng) o creae leverage. However, he acual reurns on he +x LETF are +0% for day and -0% for day, resulng n a wo-day reurn of (. x 0.9) = 8%, whch s less han wce he ndex reurn. The dfference comes from he fac ha afer he posve reurn on he frs day, he LETF ncreases s (dollar amoun) exposure o he underlyng ndex so as o manan s +x leverage rao. The hgher exposure ncreases he amoun of loss when he ndex drops on he second day. The less volale he underlyng ndex's daly reurns over he reurn pah, he hgher he compounded reurns on he LETFs wll be. Consder reurn pah #, where he ndex reurn on each of he wo days were.5% (.e., zero volaly bu wh he same wo-day compounded reurn of he ndex as n pah # ), he +x LETF's compounded reurn s 9.0%, whch exceeds wce he benchmar reurn. The same concluson obans for he case where he ndex declnes over he wo days. Consder reurn pahs # 3 and # 4. Under boh pahs, he ndex's wo-day reurn s -5.50%. However, under pah # 3 (where here s volaly), he +x LETF's compounded reurn s -%, whch s less han (.e., worse han) wce he benchmar reurn. On he oher hand, under pah # 4 (where here s zero volaly), he +x LETF's compounded reurn s -0.84%, whch s hgher han wce he ndex reurn. The effec of volaly s more pronounced n a "sdeways" mare, where he underlyng ndex moves up and down bu remans around he same level. Consder pah # 5, whch has he same reurn volaly as pahs # and #3. However, unle pah # (an "up-rendng" mare) and pah # 3 (a "down-rendng" mare), pah # 5 represens a sdeways mare wh hardly any reurn on he ndex over he -day perod. The +x LETF realzes he mos negave devaon from he saed mulple under pah # 5. Is compounded reurn s.3% lower han wce he reurn on he ndex. Gven smlar realzed volaly, a sdeways mare wll resul n a more 9

10 negave devaon from he saed mulple reurn han eher an "up-rendng" or "downrendng" mare. ex, consder he +3x LETF n Panel B. The effec of volaly and sdeways vs. rendng mare on he compounded reurns s smlar o he case of he +x LETF. However, can be seen ha he fund's hgher leverage magnfes he dfferences beween s compounded reurns and he saed mulple reurns. To presen a formal dscusson of he effec of compoundng on LETFs' reurns, consder he compounded reurn of an LETF over days:, j, j j j r, () where s he fund's leverage rao,, s he ndex reurn beween day and day, and s he fund's daly coss and expenses (ncludng any coss of fnancng and as a resul of all managemen facors). Equaon () can be rewren as: r, j, j exp ln j, j, () j j where, for smplcy, s assumed ha he fund's daly coss and expenses are consan (.e., = ). Then, a second-order Taylor expanson can be used o approxmae he rgh-hand sde of Equaon () as: r, exp s, (3) Ths approxmaon s adaped from Co (009). 0

11 where s he arhmec average of he ndex's daly reurn durng he perod, and s j, j j As r, s he sample varance of he daly reurns., s he compounded reurn on he LETF over days, one can hn of +, as he h -day value (.e., he payoff) of a $ nvesmen n he LETF. Equaon (3) saes ha he payoff s equal o he leveraged compounded reurn based on he average ndex daly reurn over he perod,, mulpled by an exponenal erm. When here s volaly n he reurn pah (.e., s > 0), he exponenal erm s less han one. The hgher he volaly, he lower wll be he exponenal erm. Therefore, gven an average benchmar reurn,, he reurn on an LETF wll be lower under more volale reurn pahs. The reurn wll be especally poor n a sdeways mare (.e., when 0). 3 There s of course an nfne number of reurn pahs ha he underlyng ndex can follow. To have an dea abou he dsrbuon of he payoff, assume ha he daly reurns on he ndex,,, are ndependenly and dencally dsrbued wh mean of and sandard devaon of. In addon, for llusrave purposes, agan assume ha he fund's daly coss and expenses s consan (.e., = ). Then, by he cenral lm heorem, he dsrbuon of + r, wll approach lognormal, as becomes large. 4 The expeced value of he payoff s. However, because a lognormal dsrbuon s posvely sewed, hs expeced payoff exceeds he medan payoff, and so wll no be wha an nvesor ypcally receves. On he oher hand, alhough here are fewer above-average payoffs, hey generally exceed he average by a subsanal amoun. In addon, snce he sewness of a lognormal dsrbuon ncreases wh volaly, hese effecs wll be even more pronounced when r benchmar reurn s volale or s large (n absolue value). r, s volale. Ths occurs when he underlyng Ths s rue regardless of wheher s posve or negave. Therefore, s possble, for example, for he holdngperod reurn on a -x bear LETF o be negave even when he benchmar's reurn over he same perod s negave. 3 Ths dscusson apples o boh leverage ETFs and nverse (.e., -x) ETFs. Inverse ETFs also rebalance her exposure daly. For example, suppose he underlyng ndex s 00 on day 0, 90 on day (.e., -0% ndex reurn from day 0 o day ), and 00 on day (.e., +.% ndex reurn from day o day ). The -day compounded reurn on he nverse ETF would be (.0 x ) = -.%, whle a shor seller would ge a reurn of zero. 4 Ths also uses he fac ha daly reurns on an LETF canno be worse han -00%. Ths s because daly adjusmens of he fund's exposure o reflec daly gans/losses have an effec of mnmzng he chances of large losses. In addon, for volale underlyng ndces, fund ssuers wll buy pu opons o hedge agans exreme negave reurns. Therefore, one plus daly reurn wll be non-negave.

12 3. Sample descrpon Our sample consss of funds racng four major underlyng ndces n he US mare he S&P 500, ASDAQ 00, Russell 000 and he Dow Jones Indusral Average. These four ndces are very well-nown o nvesors and have well-esablshed radonal (+x) ETFs wh large amouns of asses under managemen (AUM). Also, hey have an nverse (.e., -x) ETF and every possble leveraged ETFs (.e., +x, -x, +3x and -3x) on hem. Ths allows us o have a complee se of funds on whch o esmae and compare racng ably. Table dsplays he names and descrpons of he funds n our sample. In oal, here are four non-leveraged (.e., +x) ETFs and weny-sx LETFs (ncludng four nverse ETFs). As of he end of 0, he weny-sx LETFs had a combned AUM of $ bllon, whch was over 70% of he oal AUM of all equy (.e., domesc and foregn) LETFs n he US mare. 5 The larges one n he sample s ProShares Shor S&P 500 (SH) wh AUM of abou $.85 bllon, whle he smalles one s Rydex Inverse x S&P 500 (RSW) wh AUM of $36 mllon. The average sze of he LETFs n our sample s $4 mllon. Inser Table Here The las column of Table conans nformaon on funds' ne expense raos n 0. 6 These raos nclude he nvesmen advsory and managemen servce fees charged by fund ssuers, and are expressed as percenages of he funds' ne asse values. As repored, he ne expense raos for +x ETFs are all very low (rangng from 0.09% o 0.0% per year). The expense raos of LETFs are much hgher (beween 0.70% and 0.98% per year), reflecng he hgher coss n managng LETFs and he fac ha he asse base of hese funds (on whch managemen fees are calculaed) s much smaller. However, he expense raos of LETFs are n 5 As of he end of 0, here were approxmaely 60 equy leveraged ETFs raded n he US mare, wh aggregae AUMs of around $5 bllon. 6 These raos were essenally unchanged hroughou our sample perod.

13 he same range regardless of he underlyng ndces, he magnude of leverage, and wheher he funds are bull or bear funds. 4. A prelmnary nvesgaon As a prelmnary sep, we show how he funds' reurns under varous mare condons devae from he saed mulples, and hghlgh he effecs of compoundng, fnancng, and oher managemen facors on he devaons under hose dfferen mare condons. The funds' reurns are calculaed based on he changes n her AVs, no he changes n her mare prces. Ths approach s commonly used n sudes ha examne racng errors of ETFs (e.g., Frno and Gallagher, 00; Elon, Gruber, Comer and L, 00; and Gasneau, 004). I separaes he ssue of racng errors (.e., how well a fund's asse values rac he underlyng ndex) from he ssue of prcng effcency (.e., how close a fund's mare prce s o s AV). Whle ETFs' mare prces and AVs are lned by arbrage hrough he funds' creaon/redempon process, premums/dscouns can occur, especally n he case of leveraged ETFs (see, for example, Charupa and Mu, 03). By measurng reurn devaon based on changes n a fund's AVs raher han changes n s mare prces, we can focus on he effecs of compoundng, fnancng and managemen facors on LETFs' performance whle mang sure ha our resuls wll no be conamnaed by any prcng neffcency of he mare. We compue he reurn devaons (.e., racng errors) of he funds n our sample over 6- monh holdng perods sarng eher January s or July s of each year. We consder nonoverlappng holdng perods whn he common sample perod of he funds ha rac he same underlyng ndex. The number of 6-monh perods for each of our four groups of funds s herefore dcaed by he fund wh he laes lsng dae (see Table ) whn each group. For example, among he en funds followng S&P 500 Index, UPRO and SPXU are he las wo funds o be lsed on he exchange (lsng dae of June 3, 009). For hs group of funds, we herefore examne her reurns over seven 6-monh holdng perods, namely July o December 009, January o June 00, July o December 00, January o June 0, July o December 0, January o June 0, and July o December 0. 3

14 For he purpose of separang he compoundng effec from he oher facors, we use an approach smlar o he one adoped by Shum and Kang (03) and defne racng errors n wo ways: 7,, r, I,, r, j j, j, TE (4),, r, I,, r, j j, j, TE (5) where s he lengh of he holdng perod, s he fund's leverage rao,, s he ndex reurn beween day and day, and r,+ s he holdng-perod reurn on he AV of he LETF, ha s: 8 AV r,. (6) AV As defned, TE measures he dfference beween he fund's holdng-perod reurn and he mulple of he underlyng ndex's reurn over he same perod. Accordngly, TE encompasses racng errors ha can occur from managemen facors, fnancal coss and compoundng. The second erm on he rgh-hand sde of Equaon (4) s ndeed wha mos nvesors assume (ncorrecly) ha hey wll ge from nvesng n an LETF. Accordngly, TE s he defnon of racng errors ha mos nvesors have n mnd. On he oher hand, TE measures he dfference beween he fund's holdng-perod reurn and wha he fund's compounded reurn would be f here were no racng error n he daly reurns (.e., f here were no managemen facors or fnancng effecs and so he fund's reurn exacly mached he promsed reurn on every day durng he perod). In oher words, he second erm on he rgh-hand sde of Equaon (5) capures he effec of compoundng when leverage s 7 Focusng on he recen fnancal crss, Shum and Kang (03) dsenangle he effecs of compoundng and managemen facors on he performance of a sample of LETFs on commodes, domesc and nernaonal equy ndces over he perod of The AVs used n hs calculaon are adjused for dvdend and capal-gan dsrbuons ha an ETF made o s nvesors durng he sample perod. The dsrbuons are added bac o he AVs before we calculae he reurn n Equaon (6). 4

15 used. Accordngly, TE are racng errors caused by how he fund s managed and fnancng coss. The wo racng errors, TE and TE, are no he same unless = (.e., reurn over one day) or = (.e., for radonal +x ETF). Boh racng errors are expeced o ncrease wh he lengh of holdng perod, whle TE wll also be sensve o he reurn pah ha he underlyng benchmar aes durng he holdng perod. As llusraed n Secon.3, TE s expeced o be more negave n a sdeways mare and when realzed volaly s hgh. We repor n Table 3 he racng errors of he funds n our sample over 6-monh holdng perods consruced above. In columns and 3 of Table 3, we also repor he arhmec mean and sandard devaon of he daly reurns on he respecve underlyng ndces durng he correspondng 6-monh perod. Consder frs he measure TE. TE s negave for vrually all 6-monh holdng perods of all he funds. Tha s, he funds n our sample, regardless of her leverage raos, ypcally underperform her ndces based on TE. Ths s no surprsng gven ha he domnang managemen facor s he advsory and managemen servce fees, whch reduce he funds' reurns relave o her benchmar ndces. For each group of funds of each ndex, TE s much lower for he radonal (+x) ETF han for he nverse or leveraged ETFs. Ths s conssen wh he fac ha +x ETFs have he lowes ne expense raos (see Table ). Inser Table 3 Here There are a couple of noeworhy paerns n observng he TE of he hree groups of funds racng he S&P 500 ndex (Panel A), ASDAQ 00 ndex (Panel B), and he Dow Jones Indusral Average (Panel D). Frs of all, excep for SPXL and SPXS, bull LETFs have more negave TE han her bear counerpars, even hough hey have essenally he same expense raos. For example, he medan TE of SSO (a +x LETF of S&P 500, n Panel A) s -0.74%, whle ha of SDS (a -x LETF) s -0.30%. Second, agan even hough hey have essenally dencal expense raos, TE ends o be more (less) negave for bull (bear) LETFs when leverage rao ncreases. 9 For example, he medan TE for SSO (a +x LETF) s -0.74%, whle ha of UPRO (a +3x LETF) s -.07%. On he oher hand, he medan TE for SDS (a -x LETF) s -0.30%, whle ha of SPXU (a -3x LETF) s only -0.4%. 9 Bu agan, he behavor of SPXL and SPXS does no conform o hs paern. 5

16 These observed paerns can be explaned by he fnancng coss assocaed wh he generaon of he arge leverage reurn. As menoned n Secon., he fnancng effec lowers (enhances) he reurn on bull (bear) LETFs, hus mang her racng errors more (less) negave. In addon, because he fnancng effec s amplfed by he leverage of he funds, he reurns on bull (bear) LETFs wh hgher leverage are reduced (enhanced) by a larger amoun han her lower-leverage counerpars. However, he above menoned paerns are no observed for our sample of funds racng he Russell 000 ndex (Panel C). For hs group of funds, we conjecure ha, apar from he advsory/managemen servce fees and fnancng coss, oher managemen facors (e.g., ransacon coss and replcaon approach) also play mporan roles n affecng TE. Le us now urn our aenon o TE. Recall ha TE capures racng errors caused by all facors. Comparng wh TE, TE s much more volale over me. Wh respec o s magnude, TE s generally much larger han TE, whch suggess ha he compoundng effec normally domnaes all oher facors n dcang he sgn and magnude of he racng error. 0 Compoundng can maerally affec he realzed reurns on he AVs of LETFs n eher a posve or negave fashon. I can be observed from Table 3 ha, for he same underlyng ndex and n he same 6- monh perod, all LETFs have essenally he same sgn of TE regardless of wheher s a bull or bear fund. As llusraed n Secon.3, he mare condon durng he holdng perod can affec he holdng-perod reurn of LETFs. In parcular, a rendng (sdeways) mare ogeher wh low (hgh) reurn volaly resul n posve (negave) compoundng effec and hus a posve (negave) mpac on TE for boh bull and bear LETFs. Snce he compoundng effec s magnfed by he leverage of he LETF, he hgher he leverage, he larger s he mpac on TE. The swchng of he sgn of TE from perod o perod can be explaned by he dfference n mare condons from one perod o he oher. For example, among all he 6-monh perods under consderaon, all LETFs realze her mos posve TE durng he perod from July o December of 00. Ths srong posve effec can be arbued o he favorable reurn pahs (.e., a rendng mare ogeher wh a reasonably low reurn volaly) realzed by all he four underlyng ndces durng hese sx monhs (see Columns and 3). Boh bull and bear funds benefed from hs posve compoundng effec and he effec s ndeed sronger for LETFs wh hgher leverage. 0 As menoned earler, TE and TE are he same for +x ETFs. 6

17 On he conrary, a sdeways mare (.e., close o zero average reurns) ogeher wh hgh reurn volaly s he recpe for negave compoundng effec. Consder he perod from July o December of 0. Durng hs perod, all four underlyng ndces flucuaed wldly bu ended up wh only slghly negave reurns. I s herefore no surprsng ha we wness he mos negave TE for all LETFs (wheher bulls or bears) beng realzed durng hs perod. Agan, he effec s sronger for LETFs wh hgher leverage. Durng hs perod, he compoundng effec ogeher wh oher facors resuled n a reducon of 5.% n he AV reurn on SQQQ (he - 3x bear LETF on ASDAQ 00). Because of ha, he reurn on SQQQ became negave 0.64% even when he reurn on ASDAQ 00 durng he same perod was negave (see Panel B). I could be a sad surprse for an nvesor who used SQQQ o speculae on a drop n ASDAQ 00 bu gnorng he compoundng effec. The above prelmnary analyss allows us o examne a number of characerscs of he managemen, fnancng, and compoundng effecs and her mpac on holdng-perod reurns on LETFs. However, s shor of provdng he dealed nformaon one needs o judge he effecveness of he fund's managemen. A mare parcpan, who s well aware of he exsence of he compoundng effec, wll be neresed o fnd ou he answers o he followng quesons when she s selecng an LETF. Whch fund has he mos favorable (.e., leas negave) "alpha", and hus s he mos cos effecve, afer conrollng for he compoundng effec? Whch fund s able o generae he mos accurae "bea" (.e., he promsed leverage mulple) afer conrollng for he compoundng effec? An nvesor/speculaor wll be more neresed n fndng ou he answer o he former queson; whereas a hedger/arbrager mgh be more neresed n he laer. Besdes, gven he samplng error, we also need o fnd ou f he esmaed "alpha" and "bea" are sascally sgnfcan. In addon, mare parcpans may also wan o now how he "alpha" and "bea" mgh vary wh he lengh of he holdng perod. For example, here mgh be me-varyng behavor (or consrans) of he fund's managemen ha resuls n a more accurae bea over longer holdng perod han shorer holdng perod. To address hese quesons and concerns, we propose a new mehod o measure racng errors for LETFs. I nvolves regresson analyss ha allows us o conrol for he effecs of compoundng I should be noed ha alhough he averages of he daly reurns on he ASDAQ 00 and he Dow Jones Indusral Average (as repored n Table 3, Panels B and D, respecvely) are posve for hs perod, he reurns over he whole perod, whch are compounded reurns, are boh slghly negave. I s no a rare occurrence. Durng he same 6-monh perod, all of our bear leveraged ETFs generaed negave reurns even hough her underlyng ndces declned. 7

18 and fnancng on racng errors. Only by screenng ou hese effecs ha are ousde he conrol of he fund's managemen can we arrve a an accurae assessmen of he fund's managemen effcency. 5. Regresson analyss 5.. Proposed regresson echnque Sudes of radonal (+x) ETFs employ a few approaches o measure funds' racng errors, one of whch s regresson analyss where a fund's reurns (based on s AVs) s regressed on he underlyng ndex's reurns (e.g., Elon, Gruber, Comer and L, 0; Frno, Gallagher, euber and Oeomo, 004). Exsng sudes on LETFs also adop hs regresson approach where he regresson equaon s specfed as: r a b j, j e, (7), j where, as defned earler, r,+ s he -day holdng-perod reurn on he AV of an LETF,, s he ndex reurn beween day and day, and e s he resdual erm (see, for example, Lu, Wang and Zhang, 009; Charupa and Mu, 0, 03; and Shum and Kang, 03). An esmaed nercep (a) ha s close o zero ogeher wh an esmaed slope coeffcen (b) ha s close o he fund's saed leverage rao () are consdered as ndcaors of superor racng ably. On he oher hand, any sgnfcan dfference beween b and s nerpreed as he fund's nably o generae he promsed leveraged reurn. The problem wh hs convenonal approach s ha racng errors can be caused by boh compoundng and oher facors, whle he use of a sngle explanaory varable n Equaon (7) does no allow us o dsenangle he dfferen sources of errors. Consequenly, he effec of all he dfferen facors menoned n Secon can show up n boh he nercep and he slope coeffcen. Ths maes dffcul o nerpre he regresson resuls. For example, because of he compoundng effec, he esmaed value of b could be very dfferen from he fund's leverage 8

19 9 rao even f he managemen s very cos effecve and s dong a perfec job n replcang he leveraged reurn. Ths s especally rue for long holdng perods, for LETFs wh hgh leverage raos, and for bear LETFs (as opposed o bull LETFs). 3 To conrol for he compoundng effec, we propose a new regresson mehod as follows. Frs, by modelng TE as he sum of a consan (α) and a random componen (e ), we oban he followng equaon by rearrangng Equaon (5)..,, j j j e r (8) We expand he second erm on he rgh-hand sde of Equaon (8) o oban (see Appendx for deals): j j j r,,,, m m m m,,,... e,,,... (9) Unle Equaon (7), n whch only he frs-order erm of ndex reurn s consdered, Equaon (9) ncorporaes also he hgher-order erms whch ogeher capure he compoundng effec. The nercep (.e., α) and he frs slope coeffcen (.e., β) of Equaon (9) can now serve as cleaner measures of he fund's racng ably ha are free of any compoundng effec. There are alogeher - hgher-order erms n Equaon (9) wh slope coeffcens of ncreasng magnude ha s governed by he leverage rao. For example, a +x LETF has slope coeffcens of (.e., ^ - ), 6 (.e., ^3 - ), and 4 (.e., ^4 - ) for s nd-, 3rd-, and 4horder erms respecvely; whereas a -x LETF has slope coeffcens of 6 (.e., (-)^ + ), -6 (.e., (-)^3 + ), and 8 (.e., (-)^4 + ) for he same respecve erms. 3 Bear funds are subjec o sronger compoundng effec han bull funds. See Equaon (9) and he dscusson below.

20 Alhough he magnude of he coeffcen (β m - β) and he number of cross produc erms n he summaon n he square brace of each erm of Equaon (9) ncreases wh s order m, he magnude of he erm dmnshes qucly wh ncreasng order. 4 The second-order erm herefore capures mos of he compoundng effec. A more careful examnaon of he secondorder erm reveals ha measures he auocorrelaon of underlyng ndex reurns. A "rendng" ("sdeways") mare ends o have posvely (negavely) auocorrelaed reurns and hus a posve (negave) second-order erm. Togeher wh he posve slope coeffcen of he second-order erm (.e., β β > 0 regardless of wheher β s posve or negave), a "rendng" ("sdeways") mare wll herefore lely resul n posve (negave) compoundng effec on he reurns on boh bull and bear LETFs, as llusraed prevously based on he resuls repored n Table 3. 5 The slope coeffcen of he second-order erm herefore represens he exposure of LETF nvesors o auocorrelaon effec. Furher smulaon analyss (no repored) suggess ha we can gnore any erms of order hgher han he 3rd-order erm whou affecng he accuracy of racng error analyss for mos praccal purposes. 6 We herefore conduc our subsequen regresson analyss for our sample of LETFs usng only hree erms;.e., r a b j, j b,,, j b3 e,, 3, (0) 3 3 If he esmaed values of a and b are close o zero and he saed leverage rao β respecvely, we wll conclude ha he fund under consderaon s very effcen n delverng he promsed leveraged reurn. 4 oe ha when he order m ncreases, () he coeffcen s growng a he rae of β m ; () he number of cross produc erms s growng a he rae of m ; and () he magnude of each cross produc erm s growng a he rae of m. Thus, as far as he produc s smaller han uny, he magnude of he erm decreases when m ncreases. Snce s smlar o he leveraged reurn on he underlyng ndex over he holdng perod from o +, s lely o be much smaller han uny (.e., 00%) even for a one-year holdng perod. 5 oe ha β β s greaer when β s negave (.e., bear funds) han when β s posve (bull funds). As a resul, he effec of compoundng s sronger for bear funds han bull funds. 6 The resuls of he smulaon analyss are avalable from he auhors upon reques. 0

21 5.. Regresson resuls 5... Convenonal regresson approach Before presenng he resuls from our proposed regresson mehod, we conduc he regresson analyss for all he funds n our sample followng he convenonal regresson approach (.e., Equaon (7)). I serves as he benchmar when we examne he regresson resuls from our proposed mehod. We run regressons based on one-wee, one-monh, and one-quarer holdng perod, respecvely. For brevy, we presen n Table 4 only he resuls for he funds ha are based on he S&P 500 ndex. 7 In conducng he regressons, we use he common daa perod from June 3, 009 o December 3, 0 for all he 0 funds n our sample ha are racng he S&P 500 ndex. We use overlappng weely reurns o generae reurns for holdng perods of one monh (four wees) and one quarer ( wees). 8 The ewey-wes procedure s used o calculae he sandard errors of he esmaes for he regressons nvolvng overlappng observaons. 9 Inser Table 4 Here Consder, for example, he regresson esmaes for SPXU (.e., a -3x fund) for he holdng perod of one quarer. The esmaed nercep s , suggesng an average reurn shorfall of 3.3% per quarer, whle he slope coeffcen s -.495, whch s sgnfcanly dfferen from he promsed leverage rao of -3 a he % confdence level. However, he regresson does no ell us how much of he reurn shorfall and how much of he slope coeffcen devaon can be arbued o he compoundng effec. By no beng able o conrol 7 The followng llusraons and dscussons are equally applcable o funds based on he oher hree benchmar ndces. The regresson resuls for hese funds are avalable from he auhors upon reques. 8 For example, for one-quarer reurns, he frs observaon covers wee o wee, whle he second observaon covers wee o wee 3, and so on. 9 I s well nown ha he use of overlappng observaons can cause he OLS parameer esmaes o be neffcen and hypohess ess based (Hansen and Hodrc, 980). In he leraure, he ewey-wes approach s commonly used o correc hs bas. However, Harr and Brorsen (009) show ha he ewey-wes approach ends o underesmae he sandard devaons of he esmaes, and hus causes he null hypohess o be rejeced oo ofen. Ths s especally rue when he sample sze s small. Alhough our sample s no small, s possble ha he problem exss.

22 for he compoundng effec, does no faclae he comparson of cos effecveness among dfferen LETFs. For example, by only loong a he quarerly regresson resuls of Table 4, a mare parcpan, who s no fully aware of he compoundng effec, may navely conclude ha he cos effecveness and he racng ably of SPXU are much worse han hose of UPRO gven he fac ha: () he esmaed nercep of he former (.e., ) s much more negave han ha of he laer (.e., ); and () he esmaed slope coeffcen of he former (.e., -.495) s much more devaed from s leverage rao of -3 han ha of he laer (.e., 3.40) from s rao of 3. In fac, snce he compoundng effec s sronger for bear funds han bull funds of he same leverage magnude, he nercep of SPXU beng more negave han ha of UPRO does no necessarly mean ha he former was ndeed managed less effcenly. The larger reurn shorfall of he former may n fac be fully arbuable o he sronger compoundng effec ha s subjec o. We canno dsenangle he compoundng effec n conducng he convenonal regresson analyss. As subsequen analyss usng our proposed mehodology wll reveal, SPXU was acually more cos effecve han UPRO over our sample perod afer conrollng for he dfference n he compoundng effec (see dscusson below). Le us examne a par of funds wh dfferen magnude of leverage raos. From he quarerly regresson resuls presened n Table 4, he average reurn shorfall of RSW (a -x LETF of S&P 500) s.56%/quarer whch s more han wce ha of SH (he nverse ETF of S&P 500). Whou furher analyss, s hard o ell f hs poor performance of RSW s ndeed he drec resul of facors relaed o s managemen and fnancng or s smply due o he compoundng effec, whch we now s more sgnfcan for hgher-leverage LETFs. As subsequen analyss usng our proposed mehodology wll reveal, he cos effecveness of hese wo funds are acually que smlar afer we conrol for he dfference n compoundng effec (see dscusson below) Proposed regresson approach To demonsrae our proposed approach, we run he regresson of Equaon (0) for our sample of 30 funds based on one-wee, one-monh, and one-quarer holdng perod respecvely. There s no compoundng effec n daly reurns, so we do no run he regresson wh one-day holdng perod. We also do no conduc he regresson for one-year holdng perod snce he regresson resuls are no expeced o be nformave gven he possbly of mulcollneary due

23 o he hgh correlaons among he rgh-hand-sde varables of Equaon (0). To compare he regresson resuls across funds racng he same underlyng benchmar, we use common daa perod for all he funds whn each group, whch sars from he lsng dae (see Table ) of he fund mos recenly nroduced o he mare and ends on December 3, 0. As he LETFs n our sample do no have long hsory, we use overlappng weely reurns o generae reurns for holdng perods of one monh (four wees) and one quarer ( wees). Smlar o he convenonal regresson analyss conduced prevously, he ewey-wes procedure s used o calculae he sandard errors of he esmaes for he regressons nvolvng overlappng observaons. We es wheher he nercep (a) s dfferen from zero, and wheher he hree slope coeffcens (b, b, and b 3 ) are dfferen from her heorecal values of β, β - β, and β 3 - β respecvely accordng o he fund's leverage rao. 30 We presen he resuls n Table 5. Inser Table 5 Here Consder frs he esmaed nerceps. o surprsngly, all of hem are negave gven he managemen fees and expenses. Excep for SPXS and IWM, he nerceps for all holdng perods of all he funds are sascally sgnfcan (a leas a he 5% level). Comparng he nerceps of he same fund across holdng perods, he average "loss rae" due o managemen facors and fnancng effec s n general unform. For example, he esmaed nercep of he one-quarer holdng perod regresson of QLD s -0.3%, whch s approxmaely hree (welve) mes ha of he esmaed nercep of he one-monh (one-wee) holdng perod regresson of QLD (see Table 5, Panel B). In oher words, nvesors of dfferen nvesmen horzons are n fac subjec o a smlar average "loss rae" due o managemen facors and fnancng effec afer screenng ou he compoundng effec. Comparng he nerceps of dfferen LETFs for a gven holdng perod, we see ha hey have very dfferen average "loss raes". Ths s despe he fac ha all he LETFs n our sample have very smlar expense raos. I herefore suggess ha, besdes he sze of he expense rao, 30 Snce he +x ETFs do no have o rebalance her exposure daly o manan her saed mulple (and hus her holdng-perod reurns are no affeced by he compoundng effec), we do no expec he slope coeffcen of he frs erm (second and hrd erm) o devae oo far from + (0), regardless of he lenghs of he holdng perods. 3

24 oher managemen facors and he fnancng effec also come no play n affecng he degree of underperformance of he LETFs wh respec o her benchmar ndces. There are a few noeworhy paerns. Frs of all, for hose funds racng he S&P 500 ndex, ASDAQ 00 ndex, and he Dow Jones Indusral Average, bull LETFs always underperform by more han her bear counerpars. For example, he nercep erm n he quarerly reurn regresson of SSO (a +x LETF of S&P 500) s -0.3%, whle ha of SDS (a -x LETF of S&P 500) s only -0.9% (see Table 5, Panel A). Ths observaon s conssen wh he asymmerc effec of fnancng coss on bull vs. bear funds. The reurn on he former s negavely affeced by he fnancng coss, whle ha of he laer benefs from he coss. Ths effec herefore ncreases (lessens) he underperformance of bull (bear) LETFs. We however do no observe hs paern for he funds racng he Russell 000 ndex (see Panel C). I s possble ha oher managemen facors (e.g., ransacon coss and replcaon errors) mgh have domnaed he fnancng effec for hese LETFs. Second, almos whou any excepons, he hgher he leverage rao, he larger s he underperformance (.e., he more negave s he nercep erm) of he LETF ceers parbus. Ths s conssen wh he fac ha ransacon coss end o be hgher n managng LETFs wh hgher leverage raos. 3 By beng able o conrol for he compoundng effec, our proposed regresson approach allows for he drec comparson of cos effecveness among dfferen LETFs. The convenonal regresson approach, on he oher hand, does no readly lend self o such a comparson. Recall our prevous dscusson on he convenonal regresson resuls of SPXU vs. UPRO. By smply comparng he respecve esmaed nerceps ( vs ) from he convenonal (quarerly) regressons of hese wo funds as presened n Table 4, an unsophscaed mare parcpan, who s no fully aware of he compoundng effec, may navely conclude ha he cos effecveness of SPXU s much worse han ha of UPRO. We now such comparson s no far o SPXU gven he fac ha he compoundng effec s sronger for bear funds han bull funds of he same leverage magnude. The esmaed nercep of SPXU beng more negave han ha of UPRO does no necessarly mean ha he former was managed less effcenly. The larger reurn shorfall of he former may n fac be fully arbuable o he sronger compoundng effec ha 3 For bull LETFs, hs observed relaon beween leverage rao and underperformance may also be explaned by fnancng coss. Fnancng coss ncurred n generang he leverage exposure ncrease wh he leverage rao, hus furher conrbung o he underperformance of hghly leveraged bull funds. 4

25 s subjec o. Our proposed regresson approach allows us o more easly arrve a an apple-oapple comparson. Based on he quarerly regresson resuls of hese wo funds as repored n Table 5, Panel A, afer conrollng for he compoundng effec, SPXU acually has a smaller average reurn shorfall han UPRO gven s less negave esmaed nercep (.e., vs ). Our proposed approach allows us o screen ou he compoundng effec, whch could be very dfferen for dfferen LETFs, so we can arrve a he correc concluson ha SPXU was acually more cos effecve han UPRO over our sample perod. We can use he regresson resuls for LETFs of dfferen leverage raos o furher llusrae he benef of screenng ou he compoundng effec n comparng cos effecveness. Recall our prevous dscusson on he convenonal regresson resuls of RSW (a -x LETF of S&P 500) vs. SH (he nverse ETF of S&P 500). From he prevous convenonal (quarerly) regresson resuls, he average reurn shorfall of he former s more han wce ha of he laer based on her respecve esmaed nerceps. I s however hard o ell o wha exen hs poor performance of RSW s a drec resul of he sronger compoundng effec ha we expec for hgher leveraged LETFs. Wh our proposed regresson resuls, we can easly ell ha he poron of he average reurn shorfall of RSW ha s arbuable o managemen facors and fnancng effec s acually slghly smaller han ha of SH. Based on he esmaed nerceps of he quarerly regresson resuls presened n Table 5 Panel A, afer conrollng for he compoundng effec, he average reurn shorfalls are 0.3% and 0.7% per quarer for RSW and SH, respecvely. By comparng he quarerly regresson resuls for RSW presened n Tables 4 and 5, we noce ha managemen facors ogeher wh fnancng effec acually accoun for a very small fracon of he oal reurn shorfall of.56% per quarer (Table 4) for RSW. Mos of he reurn shorfall s he resul of he compoundng effec. o surprsngly, when we compare all he nercep esmaes repored n Table 4 wh he correspondng nercep esmaes n Table 5, Panel A, we fnd ha he compoundng effec conrbues more o he average reurn shorfall as he holdng perod lenghens. ex, consder he esmaes for he slope coeffcens based on our proposed regresson approach. In Table 5, even hough que a number of he frs slope coeffcen esmaes (.e., b ) are sascally dfferen from he correspondng leverage raos, mos of he devaons are no consdered o be economcally sgnfcan. In oher words, afer conrollng for he compoundng effec, he funds are n general que accurae n replcang he promsed mulple reurns. However, here are ndcaons ha he performance n leverage replcaon worsens as holdng 5

26 perod lenghens. For example, ou of he oal of 6 LETFs (.e., leveraged ETFs and four nverse ETFs), only sx esmaes of b are sascally dfferen from he saed leverage mulples for weely holdng perod, whereas 5 esmaes of b are sascally sgnfcan based on quarerly holdng perod. I seems ha he daly errors resulng from he process of generang he requred leverage are no random and hus no able o offse each oher over me. I herefore resuls n a sysemac bas n he leverage whch becomes more pronounced over long holdng perods. We noce wo oher paerns regardng b : () Bear LETFs end o devae more from her leverage raos han her bull counerpars; and () LETFs wh hgher leverage end o have b ha devae more from he promsed leverage raos. These wo observaons are conssen wh he noon ha s n general more dffcul o accuraely replcae leveraged reurns for bear LETFs and for funds wh hgh leverage raos. The generally hgher ransacon coss nvolved n hese nds of LETFs mae more dffcul for he managemens o delver he promsed reurns. By comparng he slope coeffcens repored n Table 4 and Table 5, Panel A (.e., b vs. b ), we can also quanfy he relave mporance of he compoundng effec and all oher facors n causng he devaon from he saed leverage rao. Le us examne he quarerly regresson resuls for SDS (a -x LETF of S&P 500) as an example. Wh he proposed regresson resuls, we now now ha s he compoundng effec raher han he managemen facors or he fnancng effec ha conrbues more o he devaon of he slope coeffcen (b) of (see Table 4) from he promsed mulple of -. The devaon solely due o managemen facors ogeher wh fnancng effec s found o be much smaller based on he correspondng slope coeffcen esmae (b ) of n Table 5, Panel A. By comparng he esmaes of b of Table 5, Panel A wh he correspondng esmaes of b n Table 4, we fnd ha s almos always he case ha he devaons n he slope coeffcens are predomnanly he resul of he compoundng effec raher han oher facors for holdng perod even as shor as one wee. Fnally, le us urn o he esmaed slope coeffcens of he second- and hrd-order erms (.e., b and b 3 ) of Equaon (0). Recall ha hey capure he exposure of he funds' reurns o he auocorrelaon effecs of underlyng ndex reurns. For he holdng perod of one wee, he majory of he esmaes of b and b 3 are no sascally dfferen from her heorecal values of β - β and β 3 - β respecvely. As he holdng perod lenghens o one monh and one quarer, he esmaes of b and b 3 end o devae more from her heorecal values. These resuls sugges 6