Path-dependence of Leveraged ETF Returns. Marco Avellaneda Stanley Jiang Zhang CIMS

Transcription

1 Pah-depedece of everaged ETF Reurs Marco Avellaeda aley Jiag Zhag CIM

2 ummary Review of exchage-raded fuds (ETFs) everaged ETFs 3X -3X Empirical facs abou ETFs Pah-depedece explaied Empirical validaio of he heoreical formula o 54 ETFs Rebalacig: replicaig leveraged reurs over log-erm horizos

3 Exchage-raded fuds ETF: Ivesme vehicles similar o muual fuds bu look like socks -- raded o a exchage -- radig is similar o socks (log shor margi) ETF: ca be viewed as a holdig compay or a fud -- sared as idex rackers -- acively maaged ETFs sice mid s Arbirage: auhorized paricipas ca creae or redeem ETFs i ``creaio uis -- creaio uis: 5K o K shares -- APs ofe ac as marke makers providig liquidiy

4 Milesoes 993: firs U ETF 998: firs Europea ETFs 6: firs acively maaged ETFs More facoids 989: Idex Paricipaio hares sopped by Chicago Mercaile Exchage 993: PY Trackig &P 5 (a.k.a. piders or PDR issuer: ae ree) 996: BGI creaes WEB (World Equiy Bechmark hares) laer called I-hares 998: ecor PDR rack 9 secors of he &P 5 8: 68 ETFs i U wih 6B i asses icrease of 5B i mohs 9: more ha 8 ETFs

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7 e v e r a g e d E T F s Producs offer a muliple of he daily reur of a referece idex Examples: Proshares Ulra Fiacials ETF (UYG) Offers a daily exposure o imes he Dow Joes Fiacial Idex (log % of uderlyig udex) Proshares Ulrahor Fiacials ETF (KF) Offers a daily exposure o - imes he Dow Joes Fiacial Idex (shor % of uderlyig idex)

8 Bullish leveraged ETF Auhorized Paricipas Deermie umber of ETFs ousadig daily Maager Mar of log ETF Borrow $ Ives $ i asses Marke Moey marke

9 Bearish leveraged ETF Auhorized Paricipas Deermie umber of ETFs ousadig daily Maager Mar of shor ETF hor $ i asses Ives proceeds i MM Marke Moey marke I realiy his ca be more complicaed (use of swap couerparies).

10 Hedgig wih a Toal Reur wap ETF Maager (e.g. ProFuds) oal reur swap oioal +/- % + ieres o cash wap couerparies Deermie he umber of ETFs ousadig Auhorized Paricipas cusodia Every day he fud maager eers io a oal reur swap o he referece socks (e.g. fiacial sock baske) wih a oioal equal o % of NAV * umber of shares ousadig.

11 ETFs & buy-ad hold ivesors: Cavea Empor Issues have bee raised i he markeplace peraiig o he suiabiliy of leveraged ETFs for log-erm ivesors seekig o replicae a muliple of a idex performace ``UB AG U.. brokerage busiess sopped sellig ETFs ha use leverage because such producs do o coform o is emphasis o log-erm ivesig Bloomberg News July 7 9 `` Due o he effecs of compoudig heir performace over loger periods of ime ca differ sigificaly from heir saed daily objecive. Therefore iverse ad leveraged ETFs ha are rese daily ypically are usuiable for reail ivesors who pla o hold hem loger ha oe radig sessio paricularly i volaile markes FINRA Regulaory Noice Jue 3 9

12 Char eigs cale Char Cursor OHC Values skf Rese Eer am KF/UYG Pas 3 mohs

13 Pas year Noice ha boh reurs are egaive (big) over year

14 ice icepio

15 Aoher example: FA/FAZ Direxio 3X ad -3X Fiacial ETF

16 Oil & Gas Proshares DIG (log) DUG (shor)

17 8/4/8 9/4/8 /4/8 /4/8 /4/8 /4/9 /4/9 3/4/9 4/4/9 5/4/9 6/4/9 7/4/ UYG vs. X IYF year UYG X IYF

18 8/4/8 9/4/8 /4/8 /4/8 /4/8 /4/9 /4/9 3/4/9 4/4/9 5/4/9 6/4/9 7/4/ KF vs. -X IYF KF (-)XIYF

19 ETFs: The discree model ( ) ( ) ( ) ( ) f r R R f r R R f R R N N Δ Δ + + Π + Π Δ Δ + expese raio for leveraged ETF he leveraged ETF price of he uderlyig idex or ETF price of he leveraged ETF over he h period reur of he uderlyig idex over he h period reur of

20 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Δ Δ + + Δ Δ Δ Δ + Δ Δ + + s N ds f r R f r R f r R f r R R R R R f r R f r R exp exp l l l l σ

21 Coiuous-ime model ( ) ( ) ( ) ( ) ( ) + + s ds f r d fd rd d d d d d d d d d d d d fd rd d d d dz d exp l l l l σ σ σ σ μ σ

22 Pah-depedece of ETF reurs has o do wih he volailiy ( ) r f ( ) σ s ds exp covexiy due o daily compoudig fiacig & fees volailiy I pracice we will esimae he sochasic volailiy as he -day sadard deviaio of he uderlyig ETF reurs.

23 KF vs. -*IYF 3-moh reurs covexiy volailiy Feb 8 o Mar 9 overlappig 3M reurs

24 KF vs. -* IYF 3 moh reurs volailiy Feb 8 o Mar 9 overlappig 3M reurs

25 Double everaged ETFs aalyzed i he sudy Uderlyig ETF Proshares Ulra Proshares Ulra hor Idex/ecor QQQQ QD QID Nasdaq DIA DDM DXD Dow 3 PY O D \&P5 Idex IJH MVV MZZ \&P MidCap 4 IJR AA DD \&P mall Cap 6 IWM UWM TWM Russell IWD UVG JF Russell IWF UKF FK Russell Growh IW UVU J Russell MidCap Value IWP UKW DK Russell MidCap Growh IWN UVT JH Russell Value IWO UKK KK Russell Growh IYM UYM MN Basic Maerials IYK UGE ZK Cosumer Goods IYC UCC CC Cosumer ervices IYF UYG KF Fiacials IYH RX RXD Healh Care IYJ UXI IJ Idusrials IYE DIG DUG Oil \& Gas IYR URE R Real Esae IYW ROM REW Techology IDU UPW DP Uiliies

26 Triple everaged ETFs used i his sudy Uderlyig ETF Direxio 3X Bull Direxio 3X Bear Idex/ecor IWB BGU BGZ Russell IWM TNA TZA Russell RIFIN FA FAZ Russell Fiacial ervices RG ERX ERY Russell Eergy EFA DZK DPK MCI EAFE Idex EEM EDC EDZ MCI Emergig Markes Idex

27 Tesig he Model: X ETFs /8 o 3/9 Uderlyig ETF Trackig_error(average %) sadard_deviaio (%) Ulralog Ticker QQQQ.4.47 QD DIA..78 DDM PY O IJH MVV IJR.6.7 AA IWM.6.88 UWM IWD..98 UVG IWF.5.59 UKF IW UVU IWP -..6 UKW IWN.5.9 UVT IWO.5.74 UKK IYM.44. UYM IYK..75 UGE IYC.56.4 UCC IYF UYG IYH.4.4 RX IYJ.5.74 UXI IYE DIG IYR URE IYW.5.55 ROM IDU.5.55 UPW rackig error average of arge reur - model reur where arge reur _/_ ad model reur is accordig o our formula

28 -X everaged ETFs Uderlyig ETF Trackig_error(average) sadard_deviaio ulra shor ickers QQQQ -.76%.66% QID DIA.73% 3.% DXD PY.%.74% D IJH -.%.6% MZZ IJR.8%.63% DD IWM -.7%.3% TWM IWD -.8% 3.4% JF IWF.4% 3.4% FK IW -.6% 3.3% J IWP -.3%.36% DK IWN.7%.9% JH IWO.4%.47% KK IYM -.54%.% MN IYK.9% 3.9% ZK IYC -.44%.85% CC IYF -7.96% 9.4% KF IYH -.6%.67% RXD IYJ -.8%.46% IJ IYE.43% 3.9% DUG IYR -.4% 3.3% R IYW.3%.67% REW IDU -3.7%.86% DP

29 Triple everaged ETFs sice icepio (Nov 8-Mar 9) Uderlyig Trackig_error(average) sadard_deviaio 3X bull icker IWB.44%.55% BGU IWM.8%.75% TNA RIFIN 3.67%.8% FA RG.57%.7% ERX EFA.6%.3% DZK EEM.4%.% EDC Uderlyig Trackig_error(average) sadard_deviaio 3X bear icker IWB -.8%.64% BGZ IWM.65%.76% TZA RIFIN -.63% 4.4% FAZ RG -.4%.% ERY EFA -.54%.86% DPK EEM.49%.43% EDZ

30 Trackig O (X PY) Trackig KF sice December 7 usig he acual prices ad he formula

31 Usig ETFs o arge log-erm leveraged reurs Problem: how ca ivesors replicae X he reurs of a ETF over a period > day (6m y) wih ETFs wihou udergoig rackig error? KF (-)XIYF 8/4/8 9/4/8 /4/8 /4/8 /4/8 /4/9 /4/9 3/4/9 4/4/9 5/4/9 6/4/9 7/4/9 8/4/8 9/4/8 /4/8 /4/8 /4/8 /4/9 /4/9 3/4/9 4/4/9 5/4/9 6/4/9 7/4/ UYG X IYF

32 Trea he arge reur as he payoff of a ``derivaive wrie o he ETF ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + T T r A s u A T A E e A V e du s f s r s A e payoff Fial ) ( / / σ ``Fair value of a derivaive ha delivers he arge payoff i

33 Compue Fair Value ad Dela ( ) ( ) ( ) ( ) - / / / / ) ( / / /. / ) ( V e V e A V e e A A e E e A V A A T r A T A T T r The hedge raio is proporioal o he value of he uderlig idex

34 Verificaio Cosider a dyamic porfolio ha holds Δ ETF a ime < < T. Π dollars of dπ rπ d + Δ d Δ rd Δ d + ( ) rd Δ rd rπ rπ d d + Π + Π d d Π + rd ( ) + Π rd Δrd rd Π rd dπ rπ d + Π d rd Π T rt e Π Π T e rt Π Π T T

35 /7/8 /7/8 3/7/8 4/7/8 5/7/8 6/7/8 7/7/8 8/7/8 9/7/8 /7/8 /7/8 /7/8 /7/9 /7/9 3/7/9 4/7/9 5/7/9 6/7/ Dyamically hedged KF ivesme sraegy KF (-)XIYF Redesig Rebalace sarig //8 7/7/9

36 Modified everaged ETF: A log-erm replicaio of leveraged reurs oal reur swap oioal (Dela)% ETF Maager (e.g. ProFuds) wap couerpary (e.g. G) Dela depeds o marke levels volailiy ad divided yield of m he idex Δ Π

37 Refereces Cheg Mider ad Madhava Aah The Dyamics of everaged ad Iverse-Exchage Traded Fuds (April 3 9). hp://ssr.com/absrac Avellaeda Marco ad Zhag aley Jia Pah-Depedece of everaged ETF Reurs (May 4 9). hp://ssr.com/absrac4478 No-Tradiioal ETFs FINRA Regulaory Noice (Jue 9). everaged ad Iverse ETFs: pecialized Producs wih Exra Risks for Buy-ad-Hold Ivesors EC Aler (Augus 8 9). hp:// Joae Hill ad Forser G. Udersadig Reurs Of everaged Ad Iverse Fuds (Augus 5 9) hp://