The Golden Target: Analyzing the Tracking Performance of Leveraged Gold ETFs Tim Leung∗ Brian Ward† 5 1 0 January 23, 2015 2 n a J Abstract 2 2 This paper studies the empirical tracking performance of leveraged ETFs on gold, and their price relationships with gold spot and futures. For tracking the gold spot, we find ] T that our optimized portfolios with short-term gold futures are highly effective in replicating S prices. The market-traded gold ETF (GLD) also exhibits a similar tracking performance. n. However,weshowthatleveragedgoldETFstendtounderperformtheircorrespondinglever- i aged benchmark. Moreover,the underperformance worsens over a longer holding period. In f - contrast,weillustratethatadynamicportfolioofgoldfuturestrackssignificantlybetterthan q various static portfolios. The dynamic portfolio also consistently outperforms the respective [ market-traded LETFs for different leverage ratios over multiple years. 2 v 6 Contents 7 2 1 Introduction 2 2 0 2 Gold Spot & Futures 4 . 1 2.1 Price Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0 2.2 Static Replication of Gold Spot Price. . . . . . . . . . . . . . . . . . . . . . . 7 5 1 3 Leveraged ETFs 9 : v 3.1 Empirical Leverage Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 11 i X 3.2 Static Leverage Replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 r 3.3 Dynamic Leverage Replication . . . . . . . . . . . . . . . . . . . . . . . . . . 15 a 4 Concluding Remarks 19 ∗Industrial Engineering & Operations Research (IEOR) Department, Columbia University, New York, NY 10027, email: [email protected]. Corresponding author. †Industrial Engineering & Operations Research (IEOR) Department, Columbia University, New York, NY 10027, email: [email protected]. 1 1 Introduction Goldisoftenviewedbyinvestorsasasafehavenorahedgeagainstmarketturmoils,currency depreciation,andother economicor politicalevents.1 For instance,during 2008-2009,major marketindices,includingtheDowandtheS&P500,declinedbyabout20%whilegoldprices rose from $850 to $1,100 per troy ounce. In August 2011, the price of gold reached a peak of $1,900 an ounce soon after the Standard & Poor’s downgrade of the U.S. credit rating. However,adirectinvestmentingoldbullionisdifficultformanyinvestorsduetohighstorage cost and the lack of a liquid exchange for gold bullions. In order to gain exposure to gold, investors can alternatively trade various instruments, such as gold futures, gold exchange- traded funds (ETFs), exchange-tradednotes (ETNs), and leveragedETFs/ETNs. Gold futures are exchange-traded contracts written on 100 troy ounces of gold, with a number of available delivery dates within 5 years of any given trading date. In the US, gold futures are traded at the New York Mercantile Exchange (NYMEX). The available months includethefrontthreemonths,everyFebruary,April,AugustandOctoberfallingwithinthe next 23 months, and every June and December falling within the next 72 months. Trading for any specific contractterminates on the third to last business day of the delivery month.2 Aswewillsee,thefuturespricesarehighlycorrelatedamongeachotherandthevariousgold (L)ETFs. GoldETFsandETNsaredesignedtotrackthespotpriceofgold,andareliquidlytraded onexchangeslikestocks. Infact,theSPDRGoldTrustETF(GLD),isoneofthemosttraded ETFs with an average trading volume of 6.2 million shares and market capitalization of US $33 billion as of July 2014.3 Within this gold ETF market, there are funds which seek to provideinvestorswitha returnequalto aconstantmultiple ofthe dailyreturnsofspotgold. Such funds are called leveraged ETFs (LETFs). Common leverage ratios are ±2 and ±3 and LETFs usually charge an expense fee for the service. Major issuers include ProShares, iShares, and VelocityShares (see Table 1). For example, the VelocityShares 3x Long Gold ETN (UGLD) provides a return of 3 times the gold spot price. Furthermore, one can take a bearish position on the gold spot price by investing in an LETF with a negative leverage ratio. An example is the VelocityShares 3x Inverse Gold ETN (DGLD). In this paper, we investigate the price dynamics of the various financial products related to gold described above. In Section 2, we examine through a series of regressions the price relationships between the futures contracts and spot gold. We find significant price co- movements among them, and their returns tend to be closer over a longer holding period. Furthermore,weconstructstaticportfoliosconsisting1or2futureswithdifferentmaturities to replicate the spot gold price. By comparing the tracking performance of these portfolios, we find that the 1-month futures is most useful for replicating the gold spot price. In Section 3, we discuss the price dynamics of leveraged gold ETFs, and their tracking performanceagainsttheirleveragedbenchmark. ForallgoldLETFsstudiedherein,theirav- eragereturnstendtobelowerthanthecorrespondingmultipleofthereference’sreturns,and the underperformance worsens as the holding period increases. For tracking the leveraged benchmark, we find that static replication with futures is ineffective. Therefore, we con- struct a dynamic leveraged portfolio using the 1-month futures. Over a long out-of-sample period, we demonstrate that this portfolio tracks the leveraged benchmark better than the corresponding LETFs. Amongrelatedstudies ongoldETFs,Baur(2013)examines the cost-effectivenessofgold ETFs relative to physicalgold holdings,and discuss the effect of commodity financialization on the prices of gold and associated ETFs. Ivanov (2013) uses t-tests to study the tracking 1SeeGhosh et al.(2004)andBaur and Thomas(2010)forempiricalinvestigationofgold’sroleasasafehaven. 2Historical quotes and contract specifics of gold futures are obtained from the CME Group (http://www.cmegroup.com/trading/metals/precious/gold.html). 3According to ETF Database (http://www.etfdb.com/compare/volume). 2 LETF Reference Underlying Issuer β Fee Inception GLD GOLDLNPM Gold Bullion iShares 1 0.40% 11/18/2004 UGL GOLDLNPM Gold Bullion ProShares 2 0.95% 12/01/2008 GLL GOLDLNPM Gold Bullion ProShares −2 0.95% 12/01/2008 UGLD SPGSGCP Gold Bullion VelocityShares 3 1.35% 10/17/2011 DGLD SPGSGCP Gold Bullion VelocityShares −3 1.35% 10/17/2011 Table 1: A summary of the gold LETFs, along with the unleveraged ETF (GLD). The LETFs with higher absolute leverage ratios, |β| ∈ {2,3}, tend to have higher expense fees. errors for gold, silver, and oil (unleveraged) ETFs. Using prices from March-August 2009, he concludes that such ETFs closely track their underlying assets. In contrast, we consider thetrackingperformanceofleveragedgoldETFsovermultipleyears,andconstructdynamic futures portfolio to replicate the leveraged benchmark based on spot gold. Futures are an important instrument for hedging commodities (L)ETFs as these funds are often fully or partially constructed using futures and swaps contracts rather than the physical asset. Guedj et al. (2011) provides an overview of commodity ETFs constructed withfutures. Alexander and Barbosa(2008)discusshedgingstrategieswithfuturescontracts for index ETFs and compare them against some minimum variance hedge ratios. Empirical studies by Smales (2015) and Baur (2012) show that the volatility of gold spot and futures exhibitsasymmetricresponsestomarketshocks. Thestudyarguesthatthehighersensitivity of gold volatility to positive shocks can be interpreted by the safe haven property of gold. There are a number of studies on the price dynamics of LETFs in general, including Cheng and Madhavan(2009), Avellaneda and Zhang (2010), and Jarrow(2010). They illus- trate how the return of an LETF can erode proportional to the leverage ratio as well as the realized variance of the reference index. For equity ETFs, Rompotis (2011) applies regres- sion to determine the tracking errors between ETFs and their stated benchmarks, and finds persistenceintrackingerrorsovertime. The horizoneffectisalsoillustratedintheempirical study by Murphy and Wright(2010)for commodityLETFs. Guo and Leung(2014)system- atically study the tracking errors of a large collection of commodity LETFs. They define a realized effective fee to capture how much an LETF holder effectively pays the issuer due to the fund’s tracking errors. Furthermore, Holzhauer et al. (2013) corroborate the volatility effect by usingVIX data ina linearregressionofthe returns. They alsofind thatthe change in the expected volatility is even more significant in this regression and that the volatility effect is stronger for bear LETFs than for bull LETFs. In this paper, we find a similar effect in that it is more difficult to track a negatively leveragedbenchmark than a positively leveraged one. Understanding the price dynamics and tracking performance of (L)ETFs are practically usefulfordevelopingtradingstrategies. Forinstance,Triantafyllopoulos and Montana(2011) modelthespreadbetweenmean-revertingpairsofgoldandsilverETFs,anddevelopefficient algorithms for estimating the parameters of this model for trading purposes. Dunis et al. (2013)developageneticprogrammingalgorithmfortradinggoldETFs. Leung and Li(2014) analyzetheoptimalsequentialtimingstrategiesfortradingpairsofETFsbasedongold,gold miners, or silver. Additionally, Naylor et al. (2011) find gold and silver ETFs to be highly profitable ETFs and are able to yield highly abnormal returns based on filtering strategies. 3 2 Gold Spot & Futures In this section we analyze the price dynamics of gold futures with respect to the spot. One benchmark for the spot gold price are the London Gold Fixing Indices, GOLDLNAM and GOLDLNPM. Each of these indices is only updated once per business day: 10:30 AM for GOLDLNAM, and 3:00 PM for GOLDLNPM in London times, by four members of the London Bullion Market Association (Scotia-Mocatta, Barclays Bank, HSBC, and Societe- Generale).4 Another widely used benchmark for the gold spot price is the Gold-U.S. Dollar exchange rate (XAU-USD). It indicates the U.S. dollar amount required to buy or sell one troy ounce of gold immediately. XAU-USD is frequently updated around the clock and its closing price is available for all trading days studied from 12/22/2008 through 7/14/2014. Forthesereasons,wechooseXAU-USDasourbenchmarkforthegoldspotpricethroughout this paper. Inthegoldfuturesmarket,thefrontmonths,suchasthe1-monthand2-monthcontracts, are actively traded daily. However, other available contracts are set to expire in specific calendar months within the next few years. As such, it may not always be possible to trade 6-monthand12-monthfutures,andonemayneedtoalternatewiththenearestmonth. Fora 6-monthfuturespositionweassumeapositionwhichalternatesbetween6-monthfuturesand 5-month futures and for a 12-month futures position we assume a position which alternates between 12-month futures and 11-month futures. This involves simply waiting while the 6-monthfutures (resp. 12-month)futures becomes a 5-monthfutures (resp. 11-month)after onemonthandthenrollingthepositionforward2monthsafterthesecondmonthpasses. For example, if it now January 2012 a 12-month futures contract would be the Dec-12 contract. WhenFebruary2012comesby,thisbecomesan11monthcontract,buttheJan-13contractis not available. Instead, we hold the positionas an11-monthfutures andthen in March2012, we roll the position forward into the Feb-13 contract returning it to a 12-month position. Throughout, we will use the 1, 2, 6, and 12 month gold futures contracts. 2.1 Price Dependency Webeginbyperforminglinearregressionsofthe1-dayreturnsofgoldspotversusthefutures of maturities: 1, 2, 6, and 12 months. Across all maturities, the linear relationships are all strongand they are quite similar. InTable 2, we summarize the regressionresults,including the slope,intercept, R2,androotmeansquarederror(RMSE).The R2 valuesareallgreater than 80%, indicating a strong linear fit. For every maturity, the slope is close to 0.94 and the intercept is essentially zero. The slopes suggest that the price sensitivity of futures with respect to the spot is slightly less than 1 to 1. While this may suggest that the futures prices should be be less volatile than the spot return, we find that the historical annual volatilities of the futures are higher: 19.261% (1-month), 19.263% (2-month), 19.269% (6- month), and 19.266% (12-month), as compared to the spot (18.374%). The fact that the regression results are almost the same among different futures suggests that the futures prices are highly positively correlated. Indeed, our calculations show that the correlation among the futures over the same period are all over 99%. 4According to theLondon Bullion Market Association (http://www.lbma.org.uk/pricing-and-statistics) 4 Response Slope Intercept R2 RMSE 1-Month 0.94314 2.41916·10−5 0.80947 0.00530 2-Month 0.94301 6.20316·10−5 0.80911 0.00530 6-Month 0.94348 3.17973·10−5 0.80934 0.00530 12-Month 0.94358 −5.23334·10−5 0.80984 0.00529 Table 2: A summary of the regression coefficients and measures of goodness of fit for regressing one-day returns of 1, 2, 6 and 12-month futures on 1-day returns of spot gold from 12/22/2008to 7/14/2014. The high correlation among futures prices can also be seen from their time series. In Figure 1, we plot the gold price, 1-month futures price (Jan-13 contract) and 12 month futures price (Dec-13 contract) over the period from 12/29/2012 to 1/29/2013. Over this period the 1-monthfutures and 12-monthfutures prices move in parallelto one another just asthenearperfectcorrelationwouldindicate. Furthermore,thegoldspotpriceand1-month futures price are close, as expected. On Jan 29, 2013, or trading day 21 in Figure 1, the 1-month futures would expire. We observe a slight discrepancy between the futures price and the spot on this date. While futures should theoretically converge to the spot price at maturity, in practice gold futures settle at their volume weighted average (VWAP) price withinthelastoneminute.5 Thelastpricewillbeequaltothespotpriceatmaturity,butthe settlement price used here will be calculated by weighting this last price against its volume traded and hence need not be equal to the spot gold price. Infact,thepriceco-movementamongfuturesalsohasimplicationsforthetermstructure dynamics. On a typical date in the gold market, futures prices are increasing and convex with respect to maturity as seen in Figure 2. Since the futures contracts tend to move in parallel, this leads to almost parallel shifts in the term structure. The shape of the term structure remains almostthe same overtime. During both periods we can see the increasing convex feature of the gold futures market, but in 2013 there is a reduction in convexity. In 2009, the term structure generally shifts upward from Jan to Jun, while in 2013 the term structure strictly shifts downward from Jan to Jun. 1710 12−month 1700 1−month Spot 1690 1680 Price 1670 1660 1650 1640 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Trading Days Figure1: Timeevolutionofspotgoldprice,1-monthfuturesprice(Jan-13contract) and 12 month futures price (Dec-13 contract) over the period from Dec 29, 2012 to Jan 29, 2013. We can see the two futures prices move in parallel, and that the spot price trades very close to the front month futures price. 5According to CME Group gold futures settlement procedures documentation, available at http://www.cmegroup.com/trading/metals/files/daily-settlement-procedure-gold-futures.pdf 5 980 1750 960 1700 940 1650 920 1600 Price ($)900 Price ($)1550 880 1500 Jan Jan Feb 860 Mar 1450 Feb Mar Apr Apr 840 May 1400 May Jun Jun 820 1350 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 Months Months (a) Jan-Jun 2009 (b) Jan-Jun 2013 Figure 2: Term structures from Jan to Jun in 2009 (left) and 2013 (right). Next,wecomparethelinearrelationshipsforreturnscomputedover5,10,and15trading days. Since we use disjoint time windows for return calculations, a longer holding period implies fewer datapoints for the regression. InFigure 3, we plotthe regressionsof12-month futures returnsversusgoldreturnsforboth 1-dayreturnsand10-dayreturns,plotted onthe same x-y axis scale. We can see the returns vary on a larger range for the 10-day returns. Spot gold has a 1-day return between -9.07% (4/15/2014)6 and 4.99% (1/23/2009), while its 10-day returns vary between -9.12% (6/6/2013 to 6/19/2013) and 11.30% (8/5/2011 to 8/18/2011). On the other hand, the 12-month futures has a 1-day return between -9.40% (4/15/2014)and7.68%(3/19/2009),whileits10-dayreturnsvarybetween-9.16%(6/6/2013 to 6/19/2013)and 11.30% (8/5/2011 to 8/18/2011). Moreover, we can see that the slope is slightly higher for the 10-day returns versus the 1-day returns. 0.15 0.15 0.1 0.1 s s n n ur ur et et R 0.05 R 0.05 s s e e ur ur Fut 0 Fut 0 m m − − 2 2 1 −0.05 1 −0.05 −0.1 −0.1 −0.1 −0.05 0 0.05 0.1 0.15 −0.1 −0.05 0 0.05 0.1 0.15 Gold Returns Gold Returns (a) 1-Day Returns (b) 10-Day Returns Figure 3: Linear regressions of 12-month futures returns based on 1-day returns (left) and 10-day returns (right) versus the spot gold returns. 6See http://mobile.nytimes.com/blogs/dealbook/2013/04/15/golds-plunge-shakes-confidence-in-a-haven/ 6 This is confirmed numerically and in general for the various futures contracts in Table 3. Here, we give the slopes for the regression of each futures return versus the gold return, while varying the holding period. We display the slopes and R2 values from Table 2 for comparison. However,we do not givethe intercepts for these regressionsas they are all very trivial and effectively 0. We can see that the slopes approach the value 1 as the holding period is lengthened. Thus, the longer the holding period, the more closely the gold return and futures price return are to one another. In particular, the slopes for 10-day returns are allgreaterthan1,indicating anincreasedpricesensitivity. Furthermore,the strengthofthis linear relationship increases as can be seen by the increasing R2 values in Table 3. Days 1-Month 2-Month 6-Month 12-Month Slope 1 0.94314 0.94301 0.94348 0.94358 5 0.99805 0.99752 0.99757 0.99715 10 1.01075 1.01020 1.00970 1.00841 15 0.99448 0.99461 0.99431 0.99345 R2 1 0.80947 0.80911 0.80934 0.80984 5 0.97154 0.97158 0.97194 0.97165 10 0.98516 0.98530 0.98572 0.98550 15 0.98704 0.98691 0.98725 0.98713 Table 3: A summary of the slopes and R2 from the regressions of futures returns versus gold returns over different holding periods. 2.2 Static Replication of Gold Spot Price In this section we considerreplicationofthe goldspot price with a static portfolio offutures contracts. Weuseportfoliosofeither1or2futurescontractsandaninvestmentinthemoney market account. We seek a static portfolio that minimizes the sum of squared errors: n SSE = (V −G )2, (1) j j Xj=1 where V is the dollar value of the portfolio on trading day j, while G is the dollar value of j j the gold spot price on trading day j. Letk be the number offutures contractsandw:=(w ,...,w )be the real-valuedvector 0 k of portfolio weights. In particular, w represents the weight given to the money market 0 account. To calculate the optimal portfolio value we will choose weights historically which minimize SSE over the 5-year period 12/22/2008 through 12/22/2013. Thus, we solve the following constrained least squares optimization problem: min kCw−dk2 w∈Rk+1 k (2) s.t. w =1 j Xj=0 The matrix C contains as columns, the historical prices of the various futures contracts andthemoneymarketaccount,7 andthevector,dcontainsthehistoricalpricesofspotgold. These prices are normalizedby $1000,without loss of generality,so that aninvestorstarting 7Weuse historical overnight LIBOR toconstruct an investmentin themoney market account. 7 with$1000willinvest$1000·w into the jthfutures contractand$1000·w into the money j 0 market account. We will compare our portfolios to investments in the ETF GLD, which tracks the gold spot price. To do this, we will perform an out-of-sample analysis and compare the values of $1000invested in GLD and $1000investedin our constructed portfolio overthe period from 12/23/2013to 7/14/2014. To measure the performance we will use the following root mean squared error for both in-sample and out-of-sample prices: n 1 RMSE =v (Vj −Gj)2. (3) un u Xj=1 t We solve this optimization problem for ten different portfolios with 1 or 2 futures, along with the money market account. The optimal weights, and corresponding in-sample/out-of- sampleRMSEsaregiveninTable4. Ingeneral,themoneymarketaccountisusedminimally as the weights on the account are less than 7% in absolute value for all ten portfolios. Foralltheportfolioswith2futurescontracts,theoptimalstrategyistogolongtheshorter termfuturescontractandshortthe longertermfuturescontract,withdifferentweights. The sum of the two resulting weights are approximately 1. In terms of RMSE, the 1-month futures contract appears to be the best replicating instrument of the gold spot. When it is used alone, it performs best relative to other single futures portfolios. When it is used in a pair with another futures, it performs better than any other single futures contract, and better than all other futures pairs: (2-m, 6-m), (2-m, 12-m), and (6-m, 12-m). Futures w w w RMSE (in) RMSE (out) 0 1 2 1 Futures 1-m -0.01071 1.01071 - 6.62989 3.34047 2-m -0.04835 1.04835 - 12.22148 6.12074 6-m -0.05030 1.05030 - 13.23663 5.33971 12-m -0.06842 1.06842 - 15.11103 5.71318 2 Futures 1-m, 2-m -0.00088 1.27315 -0.27227 6.26711 2.79411 1-m, 6-m -0.00171 1.23899 -0.23729 6.27232 2.97602 1-m, 12-m -0.00021 1.19336 -0.19315 6.28735 3.00006 2-m, 6-m -0.04079 5.06602 -4.02523 10.27413 9.37292 2-m, 12-m -0.01179 2.94860 -1.93681 9.65705 7.08414 6-m, 12-m 0.01481 4.80979 -3.82460 9.57938 4.52846 Table 4: A summary of the weights and in/out of sample RMSEs for portfolios of 1 and 2 futures contracts. For portfolios with 2 futures, the weight on the shorter term futures is w . For portfolios with a single futures, we havew =0. The weight 1 2 assigned to the money market account is denoted by w . 0 Inthesample,theRMSE valuesforthefutures portfoliosrangefrom6.63to15.11. Since these values are based on a $1000 investment, this means the error within the sample is between 0.663% and 1.511%, which is quite low. By comparison, our calculations give a RMSE of 2.091% for the gold ETF (GLD) during 12/22/2008 to 12/22/2013. Over this longer horizon of 5 years, our portfolios track the benchmark better than GLD. However, over the more recent, shorter out-of-sample period, 12/23/2013to 7/14/2014,GLD appears to track spot gold slightly better. The RMSE value for GLD during this period is 0.128%, whereas our best portfolio gives a RMSE of 0.279%. In Figure 4, we show the time series of the optimalstatic portfoliowith the frontmonth futures (top), and the time series for GLD. It is visible that both track the gold spot price closely over this out-of-sample period. 8 1160 1140 Portfolio Spot 1120 1100 1080 Price1060 1040 1020 1000 980 0 20 40 60 80 100 120 140 Trading Days (a) Portfolio with 1-m Futures 1160 1140 GLD Spot 1120 1100 Price11006800 1040 1020 1000 980 0 20 40 60 80 100 120 140 Trading Days (b) GLD Figure 4: Out-of-sample time series of our optimal portfolio of front month futures and money marketaccount(top) comparedto the spot price, and GLD comparedto the spot price (bottom). The trading days are over the period 12/23/2013to 7/14/2014. 3 Leveraged ETFs In this section we analyze the returns and tracking performances of various leveragedETFs. From historical prices of each LETF, we conduct an estimation of the leverage ratio, and investigate the potential deviation from the target leverage ratio. Moreover, we construct a number of static portfolios with futures contracts to seek replication of some leveraged benchmarks. However,thestaticportfoliosfailtoeffectivelytracktheleveragedbenchmarks. This motivates us to consider a dynamic portfolio with futures, which turns out to have a much better tracking performance. By design, an LETF seeks to provide a constant multiple of the daily returns of an underlying index or asset. Let us denote β ∈ {−3,−2,+2,+3} the leverage ratio stated by the LETF, and R the daily return of the underlying (gold spot). Ideally, the LETF value j on day n, denoted by L , is given by n n L =L · (1+βR ). (4) n 0 j jY=1 9 We call this the leveraged benchmark, and examine the empirical performance of various LETFs with respect to this benchmark. For many investors, one appeal of LETFs is that leverage can amplify returns when the underlying is moving in the desired direction. Mathematically, we can see this as follows. Rearranging (4) and taking the derivative of the logarithm, we have n d L R n j log = . (5) dβ (cid:18) (cid:18)L (cid:19)(cid:19) 1+βR 0 Xj=1 j With a positive leverage ratio β > 0, if R > 0 for all j, then log Ln , or equivalently the j (cid:16)L0(cid:17) valueL ,is increasinginβ. Inotherwords,whenthe underlyingassetis increasinginvalue, n alarger,positiveleverageratioispreferred. Ontheotherhand,ifR <0forallj,andβ <0, j a more negative β increases log Ln and thus L . This means that when the underlying (cid:16)L0(cid:17) n asset is decreasing in value, a more negative leverage ratio yields a higher return. The example below illustrates the consequences of maintaining a constant leverage in an environment with non-directional movements: Day ETF %-change +2x LETF %-change −2x LETF %-change 0 100 100 100 1 98 -2% 96 -4% 104 4% 2 99.96 2% 99.84 4% 99.84 -4% 3 97.96 -2% 95.85 -4% 103.83 4% 4 99.92 2% 99.68 4% 99.68 -4% 5 97.92 -2% 95.69 -4% 103.67 4% 6 99.88 2% 99.52 4% 99.52 -4% Even though the ETF records a tiny loss of 0.12% after 6 days, the +2x LETF ends up withalossof0.48%,whichisgreater(inabsolutevalue)than2timesthereturn(−0.12%)of the ETF. We can see this to be the case on any day (e.g. not just the terminal date) except for day 1. For example, on day 3, the ETF has a net loss of 2.04% and the LETF has a net loss of 4.15%, which is greater (in absolute value) than 4.08% (twice the absolute value of the return of the ETF). Furthermore, it might be intuitive that that the −2x LETF should have a positive return when the ETF and LETF have negative returns, this is not true. At the terminaldate, both the long andshortLETFs haverecordednet lossesof 0.48%. Again, this occurs throughout the period as well, not just the terminal date. In addition to day 6, both the long and short LETFs as well as the ETF itself are in the black. These results are consequences of volatility decay. Although long and short LETFs are expected to move in opposite directions daily by design, it is often possible for both LETFs to have negative cumulative returns when held over a longer horizon. Figure 5 shows the historical cumulative returns of the gold LETFs UGL(+2x)andGLL(−2x)fromJuly2013toJuly2014. Fromtradingday124(1/24/2014) onward, GLL has a negative cumulative return. There are points after trading date 124 where UGL also has a negative cumulative return. In fact, it starts in the black onthis date and continues to have a net loss until trading date 146 (2/12/2014). This occurs again a few times, another long stretch where both have a net loss is trading date 210 (5/15/2014) through233(6/18/2014).Thisobservation,thoughmaybe counter-intuitiveatfirstglance,is aconsequenceofdailyreplicationofleveragedreturns. The valueerosiontends to accelerate during periods of non-directional movements. 10