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Path-dependence of LeveragedETF ReturnsMarco AvellanedaStanley Jiang ZhangCIMS

Summary Review of exchange-traded funds (ETFs) Leveraged ETFs, 3X, -3X, Empirical facts about LETFs Path-dependence explained Empirical validation of the theoretical formula on 54 LETFs Rebalancing: replicating leveraged returns over long-term horizons

Exchange-traded fundsETF: Investment vehicles similar to mutual funds but “look like stocks”-- traded on an exchange-- trading is similar to stocks (long, short, margin)ETF: can be viewed as a holding company or a fund-- started as index trackers-- actively managed ETFs since mid 2000’sArbitrage: authorized participants can create or redeem ETFs in creation units’’-- creation units: 25K to 100K shares-- APs often act as market makers, providing liquidity

Milestones1993: first US ETF1998: first European ETFs2006: first actively managed ETFsMore factoids 1989: Index Participation Shares, stopped by Chicago Mercantile Exchange1993: SPY Tracking S&P 500 (a.k.a. Spiders or SPDRS, issuer: State Street)1996: BGI creates WEBS (World Equity Benchmark Shares), later calledI-Shares1998: Sector SPDRS track 9 sectors of the S&P 5002008: 680 ETFs in US with 610B in assets, increase of 125B in 12 months2009: more than 800 ETFs

L e v e ra g e dE T F sProducts offer a multiple of the daily return of a reference indexExamples:Proshares Ultra Financials ETF (UYG)Offers a daily exposure to 2 times the Dow Jones Financial Index(long 200% of underlying undex)Proshares UltraShort Financials ETF (SKF)Offers a daily exposure to -2 times the Dow Jones Financial Index(short 200% of underlying index)

“Bullish” leveraged ETFAuthorizedParticipantsDetermine number ofETFs outstanding, dailyManagerof longMarLETFMarketInvest 200in assetsBorrow 100Moneymarket

“Bearish” leveraged ETFAuthorizedParticipantsDetermine number ofETFs outstanding, dailyManagerof shortMarLETFMarketShort 200in assetsInvest proceeds in MMMoneymarketIn reality this can be more complicated (use of swap counterparties).

Hedging with a Total Return Swaptotal return swap,notional /- 200% interest on cashETF Manager(e.g. ProFunds)Determine thenumber of p counterpartiesEvery day, the fund managerenters into a total return swapon the reference stocks (e.g.financial stock basket), with a notionalequal to 200% of NAV * numberof shares outstanding.

LETFs & buy-and hold investors:Caveat Emptor Issues have been raised in the marketplace pertaining to the suitabilityof leveraged ETFs for long-term investors seeking to replicate a multipleof an index performance UBS AG U.S. brokerage business stopped selling ETFs that use leveragebecause such products do not conform to its emphasis on long-terminvesting’’ Bloomberg News, July 27, 2009 Due to the effects of compounding, their performance over longer periodsof time can differ significantly from their stated daily objective. Therefore,inverse and leveraged ETFs that are reset daily typically are unsuitable forretail investors who plan to hold them longer than one trading session,particularly in volatile markets’’ FINRA Regulatory Notice, June 31, 2009

Chart SettingsScale Chart Cursor OHLC Values ResetEnter namskfSKF/UYG Past 3 months

Past yearNotice that both returns are negative (big) over 1 year

Since inception

Another example: FAS/FAZDirexion 3X and -3X Financial ETF

Oil & Gas Proshares DIG (long)DUG (short)

YG vs. 2X IYF, 1 year120100806040UYG2X IYF200-20-40

KF vs. -2X IYF250200150SKF100(-2)XIYF500

LETFs: The discrete modelRS ,n return of the underlying index over the nth periodRL ,n return of the leveraged ETF over the nth periodSt price of the underlying index or ETFLt price of the leveraged ETFf expense ratio for leveraged ETFRL ,n βRS ,n (1 β )rΔt fΔtLt Π (1 RL ,n )Nn 1 Π (1 βRS ,n (1 β )rΔt fΔt )Nn 1

1ln (1 βRS ,n (1 β )rΔt fΔt ) βRS ,n (1 β )rΔt fΔt β 2 RS2,n212β ln (1 RS ,n ) βRS ,n β RS2,n ()1 2ln (1 βRS ,n (1 β )rΔt fΔt ) β ln (1 RS ,n ) (1 β )rΔt fΔt β β RS2,n2βLt St 1 exp (1 β )rt ft β 2 β2L0 S 0 β(N) Rn 12S ,n t Lt St 1 22 exp (1 β )rt ft β β σ s ds 2L0 S 0 0 ()

Continuous-time modeldS t σ t dZ t μt dtStdLtdS t β (1 β )rdt fdtLtSt2d ln Lt dLt 1 dLt dL 1 t β 2σ t2 dt Lt 2 Lt Lt 22dS t 1 dS t dS t 1 2 σ t dtd ln St St 2 St St 21d ln Lt βd ln St (1 β )rdt fdt β 2 β σ t2 dt2(β)t Lt St 1 2 exp (1 β )rt ft β β σ s2 ds L0 S 0 20 ()

Path-dependence of LETF returnshas to do with the volatilityβt Lt St 1 22 exp (1 β )rt ft β β σ s ds L0 S 0 20 convexity dueto dailycompounding(financing& fees)volatilityIn practice, we will estimate the stochastic volatility as the 10-day standarddeviation of the underlying ETF returns.

SKF vs. -2*IYF 3-month returnsconvexityvolatilityFeb 2, 2008 to Mar 2, 2009, overlapping 3M returns

Ln SKF vs. -2*Ln IYF, 3 month returnsvolatilityFeb 2, 2008 to Mar 2, 2009, overlapping 3M returns

Double Leveraged ETFs analyzed in thestudyUnderlying ETFProshares UltraProshares Ultra ShortIndex/SectorQQQQQLDQIDNasdaq 100DIADDMDXDDow 30SPYSSOSDSS\&P500 IndexIJHMVVMZZS\&P MidCap 400IJRSAASDDS\&P Small Cap 600IWMUWMTWMRussell 2000IWDUVGSJFRussell 1000IWFUKFSFKRussell 1000 GrowthIWSUVUSJLRussell MidCap ValueIWPUKWSDKRussell MidCap GrowthIWNUVTSJHRussell 2000 ValueIWOUKKSKKRussell 2000 GrowthIYMUYMSMNBasic MaterialsIYKUGESZKConsumer GoodsIYCUCCSCCConsumer ServicesIYFUYGSKFFinancialsIYHRXLRXDHealth CareIYJUXISIJIndustrialsIYEDIGDUGOil \& GasIYRURESRSReal EstateIYWROMREWTechnologyIDUUPWSDPUtilities

Triple Leveraged ETFs used in thisstudyUnderlying ETFDirexion 3X BullDirexion 3X BearIndex/SectorIWBBGUBGZRussell 1000IWMTNATZARussell 2000RIFINFASFAZRussell 1000 Financial ServicesRGSERXERYRussell 1000 EnergyEFADZKDPKMSCI EAFE IndexEEMEDCEDZMSCI Emerging Markets Index

Testing the Model: 2X ETFs, 2/2008 to3/2009Underlying ETFTracking error(average, %) standard deviation (%)Ultralong UYGRXLUXIDIGUREROMUPWtracking error average of target return - model return,where target return L t/L 0 and model return is according to our formula

-2X Leveraged ETFsUnderlying ETFTracking error(average)standard deviationultra short 67%REWIDU-3.17%2.86%SDP

Triple Leveraged ETFs, since inception(Nov 2008-Mar 2009)UnderlyingTracking error(average)standard deviation3X bull %EDCUnderlyingIWBIWMRIFINRGSEFAEEMTracking andard deviation0.64%0.76%4.04%1.01%1.86%1.43%3X bear tickerBGZTZAFAZERYDPKEDZ

Tracking SSO (2X SPY)Tracking SKF since December 2007 using the actual prices and the formula

Using LETFs to target long-termleveraged returnsProblem: how can investors replicate 2X the returns of an ETF over aperiod 1 day (6m, 1y) with LETFs without undergoing tracking error?2501201002008015060UYG2X 89/4/20088/4/20088/4/20080-40

Treat the target return as the payoff of a derivative’’ written on the LETFFinal payoff ST 1S0β t , S Lt Lt t e A(0,t ) S0 A(s, t ) (β 1)r (t s ) f (t s ) 1/ β L S t S 0 t L0 β2 β2t σs2udue A ( 0 ,t ) / β S V (Lt , At , t ) e r (T t ) E T 1 Lt , At S0 Fair value’’ of a derivativethat delivers the targetpayoff in S

Compute Fair Value and Delta r (T t ) V (L, A, t ) eE LT L01/ β eA (0.,T ) / β - 1 Lt L, A(0, t ) A 1/ β L L0 e A / β e r (T t ) 1 / β 1 L V(L, A, t ) 1 β L0 L A/ β 1e L 0L 1/ β V1 S Lβ S0 V 1 L L L β L0 e A/ βThe hedge ratiois proportionalto the value ofthe underlingindex

VerificationConsider a dynamic portfolio that holds Δ t Π 01 Stdollars ofβ S0LETF at time t , 0 t T .dΠ t rΠ t dt Δ tdLt Δ t rdtLt dSt Δt β (1 β )rdt Δ t rdt St rΠ t dt Π 0 1 St dS t β (1 β )rdt Δ t rdtβ S0 St rΠ t dt Π 0dStS1 St1 St Π 0 t rdt Π 0rdtrdt Π 0β S0β S0S0S0 dS S dΠ t rΠ t dt Π 0 t t rdt S0 S0 Π T e rT Π 0 ST e rT S 0SΠ T T S0Π 0 S0Π0

0082/7/20081/7/2008Dynamically hedged SKF investment strategyRebalance starting 01/02/08Redesign250200150100500

Modified Leveraged ETF: A long-termreplication of leveraged returnstotal return swap,notional (Delta)%ETF Manager(e.g. ProFunds)Swap counterparty(e.g. GS)Delta depends on market levels,volatility, and dividend yield ofm Stthe index Δ t Π 0β S0

ReferencesCheng, Minder and Madhavan, Ananth, The Dynamics of Leveraged and Inverse-ExchangeTraded Funds (April 23, 2009). http://ssrn.com/abstract 1393995Avellaneda, Marco and Zhang, Stanley Jian, Path-Dependence of Leveraged ETFReturns (May 14, 2009). http://ssrn.com/abstract 1404708Non-Traditional ETFs, FINRA Regulatory Notice (June notice/documents/notices/p118952.pdfLeveraged and Inverse ETFs: Specialized Products with Extra Risks forBuy-and-Hold Investors, SEC Alert (August 18, fs-alert.htmJoanne Hill and Forster, G., Understanding Returns Of LeveragedAnd Inverse Funds (August 25, rnalofindexes/

Underlying ETF Direxion 3X Bull Direxion 3X Bear Index/Sector IWB BGU BGZ Russell 1000 IWM TNA TZA Russell 2000 RIFIN FAS FAZ Russell 1000 Financial Services RGS ERX ERY Russell 1000 Energy EFA DZK DPK MSCI EAFE Index EEM EDC EDZ MSCI Emerging Markets Index