leveraged etf.pdf

EQUITY RESEARCH Equity Linked Strategies | U.S. Derivatives Strategy | November 17, 2009

Barclays Capital does and seeks to do business with companies covered in its research reports. As a result, investors should be aware that the firm may have a conflict of interest that could affect the objectivity of this report.

Investors should consider this report as only a single factor in making their investment decision.

PLEASE SEE ANALYST CERTIFICATIONS AND IMPORTANT DISCLOSURES BEGINNING ON PAGE 23.

SPECIAL REPORTS Leveraged ETPs : Myths & Reality

Although leveraged and inverse ETFs (LETFs) have grown dramatically over the past few years in AUM and trading volume, there still exists a remarkable amount of controversy and (in our opinion) misconceptions surrounding these products. In this report, we provide a comprehensive analysis and several complementary perspectives which might shed some light on the dynamics of these novel instruments.

The crucial insight we gain from our analysis is that any self-funded and non-recourse leveraged strategy is forced to have some sort of path dependence and will be adversely affected by volatility.

We generalize a formula, first derived in a previous note, relating the multi-day return of an LETF to that of its underlying. We empirically test this formula over a large universe of LETFs and provide some intuition around it.

We show that, as opposed to a naked static leveraged strategy, the equity invested in a LETF cannot become negative and thus they have an embedded put-like option. We show that this option is actually a power option and thus the hedging strategy of an LETF is equivalent to synthetically replicating this option.

Next we compare the dynamics of LETFs to a self-funded margin/futures trading strategy and demonstrate that although the payoff for this strategy is quite different, it shares the issues of path dependence seen in LETFs.

Finally, we compare LETFs with an exchange traded product which is quite popular in Europe, the so called “knock-out with stop loss” note as an alternative exchange traded vehicle to obtain leverage.

Maneesh Deshpande+1 212 526 2953 [email protected] BCI, New York

Rohit Bhatia +1 212 526 0367 [email protected] BCI, New York

Barclays Capital | Special Report

November 17, 2009 2

Introduction Leveraged and Inverse ETFs (hence-forth referred to as LETFs) have grown dramatically over the past few years in AUM and trading volume. The popularity of these products is especially remarkable given the considerable amount of controversy and attendant (in our opinion) misconceptions and general lack of understanding surrounding these products. Much of the debate and confusion revolves around the nature of the dynamics of their multi-period returns and in particular their path dependence and exposure to volatility. This has raised the specter of the suitability of holding these instruments over longer periods of time; an issue which gained prominence recently when FINRA weighed in on the issue (see here and here).

In this report, we provide a comprehensive analysis and offer several complementary perspectives which might shed some light on the dynamics of these novel instruments. The crucial insight we gain from this analysis is that any self-funded and non-recourse leveraged strategy is forced to have some sort of path dependence and will be adversely affected by volatility. The (obvious) fact that a leveraged strategy can result in negative equity means the provider of leverage will demand that the investor put in place some measures to prevent such an outcome. By definition, such measures are tantamount to buying some version of a put option which inevitably leads to a dependence on volatility.

We briefly summarize the contents and main conclusions of this report:

We begin by documenting the remarkable growth in LETFs and provide a snapshot of the current state bucketed by various criteria such as leverage, asset class and underlying index.

The stated objective of a LETF is to provide daily returns which are a multiple of those of an underlying index. We do a careful analysis of the tracking error of several LETFs and compare them to a few regular ETFs. We find that by and large these products do a respectable job in meeting their stated objectives. Besides their explicit management fee, LETFs also have several implicit costs related to the cost of leverage and hedging. The above analysis allows us to obtain an estimate of these implicit costs.

We next turn to the vexing issue of the multi-period returns for LETFs. We generalize a formula, first derived in our previous report (please see Understanding Ultra-short ETFs), which relates the multi-period return of a leveraged ETF to that of the underlying index and it’s realized volatility over that period. We empirically test this formula and find that it does an excellent job in explaining actual data for a variety of LETFs.

An important consequence of this formula is that LETFs satisfy the requirement that (in absence of large jumps in the underlying) their price always remains positive and hence they have an embedded put-like option. We show that this option is actually a power option. Instead of paying the premium upfront, the hedging strategy used by an LETF manager is equivalent to synthetically replicating this option and thus an LETF investor pays for this option “as he goes”. This is quite reminiscent of the portfolio insurance strategies popular in the 80’s which, in the absence of a robust options market, were used by investors to synthetically replicate a put option.

Of course there are other ways to mitigate the negative equity risk, the most popular being maintenance margin employed by futures exchanges and margin accounts. We show that if initial and maintenance margin were equal, the investor meets maintenance margin by reducing his exposure and extracts the maximum leverage from his futures account, his returns would exactly mimic that of a leveraged ETF. Of course, in reality, investors do not

https://live.barcap.com/go/research/content?contentPubID=FC1247695

http://www.finra.org/Investors/ProtectYourself/InvestorAlerts/MutualFunds/P119778

http://www.finra.org/Industry/Regulation/Notices/2009/P118953


November 17, 2009 3

use all their equity to increase exposure as the underlying moves in their favour. However, as we show, a “non-greedy” but self-funded futures strategy is also highly path dependent and would be hurt by volatility.

Another alternative for protection is to use a disciplined stop-loss strategy whereby once the equity in the account drops to a certain threshold the entire position is liquidated. This is institutionalized in Europe as the so called “knock-outs with stop loss” leveraged notes (hence forth referred to as KO notes). These are essentially products where the issuer lends money to the investor to gain leverage and imbeds in an automatic stop-loss. We provide a detailed analysis of the dynamics of this alternative exchange traded vehicle to gain leverage and compare them to LETFs.

Seen in this light, we believe that the debate over suitability of LETFs is somewhat misguided. The question is not whether these are suitable because of their volatility dependence which is inevitable given their inherent leverage. The real question is whether the investor likes the particular option embedded in the LETF as compared to those embedded in a regular futures strategy or a KO note strategy.


November 17, 2009 4

Leveraged ETFS: Growth and current landscape Figure 1 demonstrates the remarkable growth in LETF assets under management which

currently stands at nearly $30bn (total U.S ETF AUM ~ $700bn).

The rise in trading volumes of these products has been equally dramatic with the daily traded value of these products now exceeding $10bn/day (to put this context daily traded value for all ETFs ~ $50bn and all stocks in Russell 3000 ~$90bn). Clearly, for better or worse, this is an asset class to be reckoned with.

As shown in Figure 1 and further in Figure 3 below, while the underlying indices are still predominantly equity based, ETFs based on treasury indices have grown in popularity over the past few months (e.g. TBT the Ultra-Short 20+ Year Treasury ETF now has ~$4bn of AUM).

Figure 2 separately shows the trend in shares outstanding for long and short LETFs. It is interesting to note that the investor demand seems to be quite contrarian. While the market-sell off until Mar 09 was accompanied by increase in shares outstanding for the leveraged long ETFs, the subsequent sell-off has seen a dramatic rotation away into the short LETFs.

Finally Figures 3 and 4 bucket the LETF universe by the underlying index and leverage respectively. Clearly, as of now, the most popular structure appears to be the 2X ultra-short ETFs.

Figure 1. LETF AUM: Up, up and away Figure 2. Contrarian nature of LETF demand

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0200400600800

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es O

ut (

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)

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Source: Barclays Capital, Bloomberg Source: Barclays Capital, Bloomberg


November 17, 2009 5

Figure 3. LETF AUM bucketed by underlying index Figure 4. LETF AUM bucketed by leverage type

Underlying Index AUM ($Bn) Number

S&P 500 7.2 BarCap US 20y Treas 4.1 DJ US Financials 3.1 Russell 1000 Financial Svc 2.3 Nasdaq 100 2.2 DJ US Real Estate 1.5 Russell 2000 1.3 Dow Jones Industrial Avg 1.3 DJ US Oil&Gas 0.8 MSCI Emering Markets 0.7 Russell 1000 0.6 Others 5.2 Total 30.2

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November 17, 2009 6

Analysis Universe

For the rest of note, we will restrict ourselves to equity based LETFs which 1) have a minimum AUM of $100M, 2) have been in existence for at-least one year and 3) have a published total return version of their underlying index. The condition for the last filter will become apparent in the following section. Figure 5 shows some characteristics of the resultant universe of 32 LETFs and demonstrates that we have a fairly balanced basket of names across different underlying indices and leverage.

Figure 5: Universe of LETFs

Ticker NameUnderlying

IndexShares (x1000) AUM($) Leverage

SDS PROSHARES ULTRASHORT S&P500 SPX 84,975 3,516,266 -2

UYG PROSHARES ULTRA FINANCIALS DJUSFN 378,375 2,137,819 2

SSO PROSHARES ULTRA S&P500 SPX 52,950 1,777,532 2

SH PROSHARES SHORT S&P500 SPX 25,726 1,465,377 -1

QID PROSHARES QQQ ULTRASHORT NDX 45,675 1,081,127 -2

QLD PROSHARES ULTRA QQQ NDX 18,750 930,750 2

SKF PROSHARES ULTRASHORT FINANCI DJUSFN 33,750 885,600 -2

SRS PROSHARES ULTRASHORT REAL ES DJUSRE 86,400 869,184 -2

URE PROSHARES ULTRA REAL ESTATE DJUSRE 105,225 610,305 2

DXD PROSHARES ULTRASHORT DOW30 INDU 17,175 609,713 -2

DIG PROSHARES ULTRA OIL & GAS DJUSEN 14,700 463,491 2

DDM PROSHARES ULTRADOW30 INDU 11,850 444,968 2

UYM PROSHARES ULTRA BASIC MATERI DJUSBM 15,150 401,475 2

TZA DIREXION DLY SM CAP BEAR 3X RTY 31,400 392,500 -3

TWM PROSHARES ULTRASHORT 2000 RTY 12,900 375,648 -2

BGZ DIREXION DLY LG CAP BEAR 3X RIY 16,400 365,401 -3

DUG PROSHARES ULTRASHORT OIL & G DJUSEN 22,650 335,220 -2

DOG PROSHARES SHORT DOW30 INDU 4,575 262,285 -1

UWM PROSHARES ULTRA RUSSELL2000 RTY 8,700 229,158 2

BGU DIREXION DLY LG CAP BULL 3X RIY 4,250 209,843 3

TNA DIREXION DLY SM CAP BULL 3X RTY 4,300 190,534 3

PSQ PROSHARES SHORT QQQ NDX 3,750 182,138 -1

ROM PROSHARES ULTRA TECHNOLOGY DJUSTC 3,375 143,336 2

SMN PROSHARES ULTRASHORT BASIC M DJUSBM 11,025 124,803 -2

RSU RYDEX 2X S&P 500 ETF SPX 4,150 119,271 2

RWM PROSHARES SHORT RUSSELL2000 RTY 2,400 113,424 -1

RSW RYDEX INVERSE 2X S&P 500 ETF SPX 1,500 100,020 -2

SEF PROSHARES SHORT FINANCIALS DJUSFN 2,175 99,680 -1

USD PROSHARES ULTRA SEMICONDUCTO DJUSSC 3,375 93,960 2

SCC PROSHARES ULTRASHORT CONS SV DJUSCY 1,350 61,668 -2

REW PROSHARES ULTRASHORT TECHNOL DJUSTC 900 26,046 -2

SIJ PROSHARES ULTRASHORT INDUSTR DJUSIN 825 23,826 -2

Source: Barclays Capital


November 17, 2009 7

Daily tracking error of LETFs: Explicit and implicit costs In this section, we examine how well the leveraged ETFs meet their stated objectives of replicating a multiple of the daily return of the underlying index. As we shall see below, this is actually a non-trivial exercise which needs to be done with some care.

To begin with, we caution that it is incorrect to simply compare the price returns of the ETF with those of the underlying index. Firstly, as with any ETF, one should use the closing net asset values (NAV) of the ETF instead of the ETF close prices. The actual closing price of an ETF will be affected by supply and demand at the close and is unlikely in general to match the NAV published by the ETF manager. Secondly, one must compare total returns and not the price returns of the ETF and the underlying index since both could pay dividends. Finally, the standard “fees and expenses” clause, which appears in any ETF prospectus, is not innocuous for leveraged ETFs since they will have systematic costs in addition to the management fees. These implicit costs are directly associated with the costs of obtaining leverage.

In order to make the above discussion concrete, consider the hedging strategy for replicating an m times levered ETF with a current NAV of $100. We first assume that m is positive. Then,

Since the ETF manager is required to have an exposure of 100*m, he must borrow 100*(m-1) worth of cash which would result in a total cost of 100*(m-1) *( Srf + )

assuming that the manager gets charged a spread of S over some reference interest, fr

We assume that the ETF manager will lend these securities out to other investors looking to short these securities. In general, the underlying security might have additional borrow cost, which we denote by b which would result in a rebate (gain) of the entire 100*m*b to the ETF manager.

Any dividends paid by the underlying index ( Iq annualized yield) will result in a positive

cash flow of 100*m* Iq . On the other hand, the dividends paid out by the leveraged ETF

to the investor ( Lq annualized yield) will result in a negative cash flow of 100* Iq for

the fund.

Finally, the management/expense fee (denoted by f ), will result in a negative cash flow of 100*f from the ETF.

Now consider a manager of a short LETF. He shorts 100*m worth of the underlying and receives cash proceeds for that amount. He now has a total of (1-m)*100 cash (note that m is negative) which he lends out at an interest rate, Srf − , which represents a gain of

( Srf − ) (1-m)*10. On the other hand, the underlying index dividends and the extra

borrow now become a costs which can still be written as 100*m*b and 100*m* Iq

respectively. The ETF dividends and management/performance fees continue to be costs.

Putting all these factors together, the daily NAV price return of the leveraged ETFL

Pr is

related to the price return of the underlying index I

Pr by the following equation:

( ) ( ) tftqtSmtrmtbqmmrr Lf

IIP

LP Δ−Δ−Δ−−Δ−−Δ++= 11

Using, total returns, we can simplify the above equation to the following form:


November 17, 2009 8

( )( ) ( ) tftrmmrtfSmrmmbmrr fI

fIL Δ−Δ−−=Δ−−−−−+= *111

Here, Smmbff 1* −+−= represents the daily “effective cost” borne by an LETF

investor.

The above discussion implies that the job of the leveraged ETF manager is quite non-trivial:

Firstly, similar to a regular ETF, he needs to maintain a portfolio replicating the index (which depending on the underlying index can be quite a non-trivial task).

Secondly, the size of the hedge needs to be rebalanced at the close everyday to maintain constant leverage. Since the precise exposure required is only known at the close but the rebalancing decisions need to be made prior to the close, any volatility around the close can result in an imperfect hedge portfolio.

Finally, he also needs to optimally and actively manage his funding and borrow risk either by rolling on a daily basis or by locking in his funding level on a longer term basis as is being anecdotally being done by some ETF managers.

We now empirically evaluate how well the above relationships hold in practice. We first calculate the total NAV returns and regress them against the total returns of the underlying indices (obtained from the published total return versions). In general, we find that the resultant slopes are in excellent agreement (to the third decimal place) with leverage value. Figure 6 shows the relationship for three leveraged ETFs and SPY for comparison.

Figure 6. Daily tracking errors of LETFs Figure 7. Annualized implicit costs for LETFs

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SDS UYG SSO QID


We then calculate the daily values of *f by using the overnight LIBOR for the reference

interest rate, fr . The mean value of *f over some time period thus represents the total

extra cost or tracking error borne by the LETF investor. Figure 7 plots the cumulative monthly values for a few representative LETFs. Clearly there is substantial variation across time especially during the more stressful 4Q08. Figure 8 shows the the YTD costs across our universe of 32 LETFs where we have subtracted the management fees/expense ratios


November 17, 2009 9

(which are typically of the order of 95 bps for LETFs) and normalized by the amount of absolute leverage. We have also included three non-leveraged ETFs (SPY, IWM and QQQQ) as benchmarks. The extra costs for SPY and IWM are negligible and are actually negative for IWM (the Russell 2000 ETF) due to the rebate the fund gets due to lending out their inventory. While there is wide variability across our LETF universe, these results indicate that in addition to the explicit management fees/expense ratios these instruments also incur non-trivial implicit costs.

Figure 8: YTD annualized costs for ETFs (normalized by leverage and net of management fees and interest rate costs)

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Multiple-day Performance of Leveraged ETFs In this section we turn to the issue of the multi-day returns for LETFs. One of the most enduring and persistent sources of confusion among investors is the apparent paradox that the long term performance of an ultra-short ETF can be negative despite a negative performance of its underlying index. As shown in Figure 9, the investor experience with the Ultra-short Real Estate ETF (Ticker: SRS) during 4Q08 is a particularly egregious example. For example from the end of Sep 08 to the end of Dec 08, the underlying Dow Jones US Real Estate Index (Ticker: DJUSRE fell by ~41% and at the same time, SRS, despite seeking to provide a leveraged short exposure itself dropped by ~18%.

More generally, as shown in Figure 10, the dependence of the multi-period returns of an LETF on those of its underlying is not straightforward. In our previous note we had derived a formula which shed some light on this issue. In this section we extend this formula to include the effects discussed in the previous section and also provide strong empirical evidence of its veracity.


November 17, 2009 10

Figure 9. DJUSRE drops and so does SRS… Figure 10. Multi-day returns of SRS are not -2X underlying index

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The “Golden Rule” for Leveraged ETFs Before proceeding, we high-light two important features stand out from Figure 10. Firstly, while SKF return is certainly correlated with the DJUSFN return, the relationship is not deterministic since the same underlying return can lead to quite different LETF returns. Secondly, the general trend appears to be quite non-linear and certainly nowhere close to a simple inverse 2X linear relationship that one might naively expect.

Our original derivation assumed that the underlying follows a simple geometric Brownian motion and also implicitly assumed a zero interest rate. Incorporating the effect of the interest rate is easy since it only changes the drift term and we get:

( ) ( ) TfTrmTmmmIE

feeRR*

2

)1(21

11 −−−−−+=+

σ

Here ER and IR are the total return of the LETF and the underlying index respectively over

a time period T and σ is the volatility of the underlying index. While the above equation was derived for a simple geometric Brownian motion, it can be shown that it is valid for a general stochastic process if one simply replaces σ with the future realized volatility over the path of the underlying. We emphasize that this relationship is probably the most important equation to keep in mind while using leveraged ETFs and should rightfully be called the “golden rule” for leveraged ETFs.

We can get some intuition around the above by looking at some extreme path scenarios:

Underlying index is completely unchanged: In this case the volatilityσ is zero and thus the first two terms are both equal to 1. We are still left with the last term which is nothing but compounded version of the daily cost of holding the leveraged position discussed in the previous section.



Underlying index moves up by r% every day: Once again the volatility is zero. The final

value of the underlying is then ( )Tr+1 and that of the LETF is ( )Tmr+1 . For large T,

these become the familiar continuously compounded expressions given by rTe and mrTe respectively. This explains the power law behaviour of the first term in the above

equation.

Underlying index’s log returns are constant but reverse each alternate day: In this case, the total return of the underlying is zero. However, it is clear that the total return of the ETF will not zero since the up and down moves of the ETF will not precisely cancel out.

While the convexity term does result in a drag in performance, it is very important to realize that this is a drag relative to a power law base and not relative to a mX linear base. This is illustrated in Figure 11 where we compare the returns of a leveraged ETF to that of a power law and a simple 2x line corresponding to a static leveraged position. Figure 12 simply plots the same data but this time now relative to the static leverage base. Thus while for small moves the LETF underperforms the static leverage position it will outperform for large moves. The probability weighted out-performance is however zero. In this sense, LETFs are fair instruments.

Figure 10. Convexity of an LETF Figure 11. Convexity of an LETF

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The payoff in Figure 12 behaviour is quite reminiscent of the payoff of a long straddle position. The upfront premium that needs to be paid to put on a straddle imposes a break-even move which the underlying must move by before the position becomes profitable. The convexity term in the golden rule thus appears to be equivalent to some kind of option premium. In the following section, we will make this analogy more explicit.

Empirically testing the Golden rule Note that, for fixed volatility, there is no path dependence implied by the above equation since every path will have the same realized volatility. The different returns for SRS for the same return of the underlying are purely because the underling volatility varied dramatically during this period. The non-linearity in Figure 10 is simply a consequence of the power-law dependence in the golden rule. This can be explicitly verified by calculating the theoretical



returns for each 3M period and comparing it to the actual return as shown in Figure 11. The slope of this line is very close to 1 as would be predicted by the golden rule.

Figure 12: YTD annualized tracking error for ETFs (normalized by leverage)

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Theoretical SRS 3M Return

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Source: Barclays Capital, Bloomberg

We carried out a more comprehensive empirical test across our selected universe of 32 LETFs. We can recast the golden rule as:

( ) ( ) fIE rmbTmmbRmbR )1(2

11log)1log( 3

2

21 −−−−+=+σ

Testing the golden rule then is equivalent to testing if the coefficients 1b and 2b equal 1.

Figure 12 shows the values of these coefficients and clearly barring a few exceptions all the slopes are very close to the predicted values of unity.

Figure 13: Testing the Golden Rule for the entire universe of stocks

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RSWDDM

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Regression coefficient b1 Regression Coefficient b2

Source: Barclays Capital,Bloomberg



LETFs as power options In this section, we explicitly characterize the optionality inherent in LETFs.

Consider a zero-strike power call option whose payoff at time T is given the m’th power of the price of the underlying:

m

TT PX =

The current value of this option is then simply given by the risk neutral expectation:

mT

rT PeX −=0

Assuming that the underlying follows a Black Scholes process, this can be explicitly evaluated to give:

rTmTmmm eePX )1()1(00

221 −−= σ

Thus we can write:

mTT PPXX )/(/ 00 =

rTmTmm ee )1()1( 221 −−−− σ

, which is precisely the golden rule! In other words, in a Black-Scholes world, an LETF is a zero-strike power call option where the premium is paid at maturity. Next let us calculate the delta of the above option:

rTmTmmm eemPdPdX )1()1(1000

221

/ −−−= σ

This can be re-written as:

0000 // PmdPXdX =

This is exactly equivalent to the relationship between the returns of the leveraged ETF and the underlying. In other words, the LETF manager (seeking to maintain constant leverage) is effect synthetically replicating a power option with the amount of the underlying he holds being what is required to be keep the option position delta neutral.

But what is the premium that the investor pays for being long this option? If the investor had asked an option market maker to make a price for this option, the dealer would have come up with a price using the implied volatility being priced into the vanilla option market. The market maker would then delta-hedge this option and any discrepancy between the implied and actual realized volatility would result in either a profit or loss for him. By synthetically replicating the option, the investor in effect is taking the risk borne by the market maker. The premium he effectively pays is determined by the actual realized volatility of the underlying and thus instead of being a fixed quantity is itself unknown at outset. The fact that delta of the power option is independent of the time to expiry means that the ETF manager does not need to concern himself with the investors holding period T.

The situation is analogous to the concept of “portfolio insurance” popular in the 80s. In the absence of a robust option market, investors who wanted to be long puts had to synthetically replicate it themselves by doing delta hedging.



The above discussion indicates that the convexity “drag” term in LETFs is simply the implicit cost that the investor pays for the optionality inherent in these products.

However, there is an important consideration to bear in mind. As the size of leveraged ETFs indexed to an underlying index increases, the hedging requirements will themselves impact the price and thus increase the realized volatility. As a result, the investors implicit cost will also increase correspondingly.

Since, the power option payoff is unfamiliar to most investors; we next calculate the equivalent number of vanilla calls and puts by using a procedure similar to that used to replicate variance swaps. This procedure is equivalent to doing a piece wise linear approximation of the non-linear payoff function. Figure 8, shows the result for double long and inverse ETF. The leveraged long ETF is equivalent to holding an equal number of out-of-the money options for each strike while the ultra-short inverse ETF requires an increasing number of lower strike options.

Figure 14: Basket of options replicating LETFs

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Thus an investor choosing to invest in an LETF needs to decide if he would like the payoff profile associated with the above basket of options. The non-trivial aspect of this is of course that the premium he would pay for these options would depend on the actual realized volatility over his holding period.

The above discussion assumed that the underlying follows a simple geometric Brownian motion. While the discussion for a more general stochastic process is beyond the scope of this note, in our opinion, the basic intuition of the results discussed here still work.

To summarize, leveraged ETFs can be viewed as synthetically replicated zero-strike power call options. However, instead of paying an upfront cost for the option, the investor pays for the option “as he goes” depending on the actual realized volatility of the underlying.



LETFs and margin based hedging strategies The motivation behind the creation of leveraged ETFs was to create an exchange traded vehicle for investors to gain leveraged exposure to an underlying. How did we end up with a power option like payoff? As discussed in the introduction, the answer is related to the (obvious) fact that a static leveraged position, by definition, can lead to negative equity. Consider a “static leveraged” position where the investor buys (m-1) times his equity to gain exposure to m times the index. The problem is that for a 100/m percent move for the underlying index the loss for the investor exceeds his original equity.

As is well known, the typical strategy to mitigate this risk used by futures exchanges and margin account providers is via maintenance margin whereby as the underlying moves in an adverse direction the investor is forced to put in additional money into his account. The problem is that if more money is put in, the equity in play changes dynamically making the entire concept of return ambiguous. A simple way to facilitate a fair comparison with leveraged ETFs where the initial equity is fixed is to assume that the investor reduces his position to raise funds to meet margin calls.

For simplicity assume that the maintenance and initial margin are equal. Then amount of initial margin simply imposes the maximum amount of leverage that an investor is allowed. If the underlying moves in an adverse direction, his current leverage will increase and the act of reducing his position (or putting in additional equity) will bring his leverage back to original level. This is exactly the same as what is done by the LETF manager. On the other hand, if the underlying moves in a favourable direction, the investor’s equity is now more than is required by the margin rule. If he does nothing, his current leverage will decrease. However, suppose that the investor decides to use the excess equity in his account to increase his exposure to the maximum amount allowed. Once he does this, once again his leverage is back up to the original level similar to that of an LETF. Thus the rebalancing process employed by the LETF manager is exactly equivalent to a self-funded and maximally leveraged futures trading strategy.

Of course, a typical futures or margin-based investor is unlikely to be as aggressive in monetizing his positive equity as the underlying moves in favourable direction even if he is forced to reduce if the underlying moves in the opposite direction. We next investigate the performance of such a non-aggressive “normal” but self-funded strategy. To be explicit: if the underlying moves in a favourable direction, the investor does nothing and allows his effective leverage to decrease. However, if the underlying makes a new low (high) for a bullish (bearish) position respectively, his effective leverage would become higher than his initial leverage. The investor then reduces his position to bring his leverage back to the initial level.

For zero interest rate, we can write down the following simple formula for the terminal value for such a strategy:

m

worst

worst

tt

PPm

PPm

FF

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

00

1

Here worstP is the “worst” value attained by the underlying and would thus correspond to

the minimum and maximum underlying price for a bullish and bearish position respectively. Thus, for a bullish position, if the underlying price never drops below the initial price the return would be same as the static leverage position which corresponds to the first factor



above. Note that the volatility of the underlying does not matter as long as it does not drop below the initial price. On the other hand, if the underlying were to go down in a straight line, the return would be same as that of a leveraged ETF with zero volatility which explains the second factor. If the underlying goes down in a straight line and then rallies back, the total return is then a product of the two factors since the value of the position before to the rally is the second factor. What happens for a general path where the underlying fluctuates around, sets a minimum and then settles at some terminal price? Interestingly, for zero interest rates, the above formula still holds. The key point to remember is that the only time any re-hedging is done is on the down moves and so even if the underlying does not go down in a straight line, the hedging strategy is the same as if it had gone down in a straight line. Non-zero interest rates make things a little complicated since the total cost will depend on the precise timing of the down moves.

Thus as expected, the self-funded futures trading strategy is also path dependent. While for leveraged ETFs we need to know the realized volatility of the underlying over the path, for

self-funded futures we need to know its worst value. Since the expected value of worstP for

a given tP will worsen as volatility increases, higher volatility will hurt performance.

We next do a Monte Carlo simulation to gain some insight and illustrate the above discussion. We assume that the underlying follows a simple geometric Brownian motion with volatility of 50%, the drift and interest rates are zero and the investment horizon is six months. We assume that the maximum leverage that investor is allowed is 3x and simulate both a 3x long LETF and a normal futures trading strategy. The results are shown in Figure 15 and 16.

The LETF performance follows the (hopefully by now) familiar golden rule. As discussed above for lower underlying prices, the performance of the self funded futures strategy is very similar to that of the LETF. On the other hand, for higher prices, the futures trading strategy clearly underperforms the LETF since it cannot do much better than a simple 3X return. The cloud of points below this line, correspond to the case when the underlying spent some time below the starting price and then ultimately went higher. Since the investor is forced to reduce his position during the down-turn his participation in the upside move is correspondingly lower. While this behaviour does lead to underperformance for large positive moves, for small moves, the futures strategy actually outperforms the LETF. Thus for example if the underlying were to initially go up by $10 and then down by $10 over the simulation horizon, the futures strategy would flat but the LETF would lose money. We also note that the futures trading strategy is much more path dependent with scatter in the returns for a given underlying price being much larger than that of the LETF. Since we have assumed a constant volatility in our simulation, the LETF payoff very closely follows the theoretical golden line and the scatter we see is purely because the actual realized volatility during each simulation run is not always equal to 50%.

Thus we see that while both these strategies behave similarly for low underlying prices, they behave quite differently for current or higher prices. However, the important point to note is that both these strategies are completely “fair” in the sense the probability weighed performance of these strategies is identical.



Figure 15. Self funded futures vs LETFs Figure 16. Self funded futures vs LETFs

0

100

200

300

400

500

600

700

800

19 50 79 110 140 171 201

Underlying Price

Pric

e

3X Self-funded Futures 3X Long Leveraged ETF

-3

-2

-1

0

1

2

3

4

19 50 79 110 140 171 201

Underlying Price

Ret

urn

vs S

tati

c Le

vera

ge

3X Self-funded Futures 3X Leveraged ETF


To summarize: investing in leveraged ETFs is equivalent to investing in a self-funded fully invested futures trading strategy. Seen in this context, the volatility dependence of leveraged ETFs is not surprising and is completely a consequence of protection.

Knockout Notes: An Alternative for exchange traded leverage While prudent margining is a one way that the counterparty to a leveraged transaction can manage its risks, another approach is to impose a stop-loss wherein the position is automatically closed once the equity value breaches a certain level. As discussed in the introduction, this approach has been institutionalized in Europe in the so called “knock-outs with stop loss” market which we now discuss in more detail.

According to Swiss Structured Products Association, as of July 09 roughly CHB 238bn was held in structured products within bank client accounts and out of these leveraged products account for the bulk of new issuance (83% in September). The leverage products category includes the following sub-categories (with the number of products listed in parenthesis): Warrants (88,525), Knock outs with stop loss (36,144), Knock Outs without stop loss (17,671), Exotic warrants (1,886). Thus purely in number of products around 25% of the total leveraged products is invested in “knock-outs with stop loss”. Assuming that the money invested is roughly proportional to the number of products we arrive at a crude estimate of ~ $40bn as the size of these products. Coincidently this is quite comparable to the current $30bn size of the LETF AUM.

What is a KO note? A KO note is an enhanced return note where the investor buys or sells an asset using leverage with an imbedded stop loss. Essentially, the issuer makes a loan to the investor which along with the initial equity is used to buy/sell a leveraged amount of the underlying. These notes are thus equivalent to a futures or a margin position with a stop-loss order.



Typically, the interest on the loan accrues at a rate which resets daily and is typically a spread over some benchmark interest rate. Formally, the value of an m-times leveraged KO

note with an issue price 0M and a stop loss level SLα at any future time T can be written

as:

rTTT emSSm

MM −−−= )1(

00

if TsMM

SLs ≤> ;0

α

)(

00

sTrsT eMM

MM −= if Ts

MM

SLs ≤<= ;0

α

,where for simplicity we have assumed a constant interest rate and 0M is the value of KO

note at inception and also assumed that if the stop-loss level is hit, the proceeds are reinvested at the interest rate, r.

Note that the financing charge always accrues and as a result the security loses value with time even if the underlying asset price is unchanged. Conversely, for a short position, while the value of the KO note will typically increase with time the opposite can happen if the borrow costs increase the value could actually decrease. As discussed above, this feature is of course also present for LETFs and is simply a consequence of obtaining leverage. However, while the interest rate cost entered as a compounded multiplier for LETFs, here it enters as an additive factor. The reason for this is that the amount of cash per share borrowed by the investor in KO notes is constant and only determined by the leverage at issue date. On the other hand, the LETF investor increases (decreases) the amount of money he borrows as the underlying moves in a favourable (adverse) direction.

Thus if the investor buys the KO note at issue date, other than the funding costs, his returns are exactly equal to m times the return of the underling as long as the stop-loss level is not breached. In other words, even for multi-period returns he has constant leverage.

What happens if the investor buys the note at some later time t? It is important to note that as opposed to LETFs, the KO notes have an explicit dependence on their inception date. Thus the absolute amount of money that the investor can borrow/lend is always equal to what he would have obtained at issue date (net of interest costs). However, if the underlying index value has moved in a favourable (adverse) direction the amount of equity that the investor needs to invest is correspondingly lower (higher). This also means that the “effective” leverage in the note for a new investor also changes. We can write the expression for value of the KO note at time T as:

)(** )1( tTr

t

T

t

T emSSm

MM −−−−= if Ts

MM

SLt

s ≤<= ;*α

Here *m and *SLα are the effective leverage and stop-loss level given by the expression:

=*m rtt

t

emSSmSSm

−−− )1()/()/(

0

0 , t

SLSL MM 0* αα =

The expression for the effective leverage is simply the ratio of exposure to the underlying to the net equity. Thus, if the investor buys the KO note at a later date, the qualitative behaviour is the same but with different leverage and stop-loss ratio.



Figure 17: Effective leverage of a KO note

0.00

5.00

10.00

15.00

20.00

25.00

0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00

Underlying Return

Effe

ctiv

e Le

vera

ge

m=2 m=3


As the underlying price moves away from its value at issuance, the effective leverage also

changes. Figure 17, plots the dependence of *m on stock price for the case of a double long and double short KO note. Clearly, the effective leverage decreases as the underlying moves in a favourable direction. This would potentially decrease the appeal of this product in the secondary market and the issuer typically issues a new note once that happens and/or retires the note. On the other hand, for adverse movement in the underlying, the effective leverage will increase but is ultimately capped by the presence of the stop-loss feature. As discussed above, although the dollar value of fresh equity that an investor invests per share obviously depends on the price of the underlying, the dollar amount of money being lent to him by the issuer is always the same. In addition, even for fixed underlying price, the effective leverage changes with time due to the explicit dependence time dependence in the denominator. Similarly, as the underlying moves in an adverse (favourable) direction, the effective stop loss ratio also increases (decreases) respectively.

To summarize, the future price dynamics of leveraged ETFs at any given point in time is exactly the same and has no explicit dependence on the current time and price of the underlying. On the other hand, the KO note behaviour changes with time and price of the underlying. In general, for increasingly adverse moves, the product becomes more risky for a new investor. The upside for this changing behaviour is of course the simplicity of future returns.

We now explicitly simulate the returns of the KO note with that of a leveraged ETF using the same parameters we used in the previous section. To be specific we consider a KO note whose leverage at inception is 3X and the stop-loss level is at 10% of the note’s issue price. The results are shown in Figures 18 and 19.



Figure 18. KO notes returns vs LETFs Figure 19. KO notes returns vs LETFs

-100

0

100

200

300

400

500

600

700

800

19 50 79 110 140 171 201

Underlying Price

Pric

e

3X KO Note 3X Long Leveraged ETF

-4

-3

-2

-1

0

1

2

3

4

19 50 79 110 140 171 201

Underlying Price

Ret

urn

vs S

tati

c Le

vera

ge

3X KO Note 3X Leveraged ETF


As should be intuitively obvious, the two branches of the KO note performance correspond to the case when the stop-loss level is breached and not breached. As long as the stop loss level is not breached, the performance of the KO note is exactly the same as a static leveraged position. On the other hand, once the level is breached, the investor has no more exposure and thus if the underlying were to subsequently rebound he will underperform relative to both a leveraged ETF and a self-funded futures strategy discussed in the previous section. Thus with both leveraged and self-funded futures strategies, the investor always has some “skin” in the game. His exposure decreases as underlying moves in adverse direction but never goes to zero. For KO notes, the exposure remains constant but gets completely knocked-out once the stop-loss level is reached. The number of points in the second branch line is simply determined by the probability that the stop-loss level will be breached.

For LETFs, the investor needs to have a view on volatility of the underlying. On the other hand in the case of a KO note the investor only needs to have a view on the likelihood that the stop-loss barrier will be breached. While the probability of the underlying breaching will increase with increasing expected volatility, the payoff for the KO note is not directly to the daily path volatility of the underlying.

Much of the controversy around LETFs is centred on the potential impact of their daily rebalancing at market close. As discussed above, LETF need to be rebalanced daily in order to maintain constant leverage. In addition, for favourable moves in the underlying index, the amount of the hedge increases. In contrast, as long as the stop-loss level is not breached, the hedge for the KO note is completely static. Of course, if the stop-loss is breached the entire hedge needs to be unwound.



Conclusions From the above considerations, we are in a much better position to answer the question about whether leveraged ETFs are “suitable” as a buy and hold instruments. The key insight gained from the analysis in this note is that some sort of path dependence is unavoidable given the need for protection in any leveraged strategy. So a recommendation that these products are unsuitable is tantamount to saying that any leveraged strategy is unsuitable. In our opinion, the right question is: what kind of path dependence does the investor want? We have examined three different leveraged strategies with quite different characteristics and we now compare them along various dimensions in Figure 20.

Figure 20: Comparison of different leveraged strategies

Leveraged/Inverse ETFs Self-funded futures KO notes

Negative Equity Protection Yes Yes Yes

Effective Leverage Stays Constant Decreases(stays constant) for favorable(adverse) move in underlying

Decreases(increases) for favorable(adverse) move in underlying

Effective Leverage Constant Variable Variable

One day return Effective leverage * Underlying Return

Effective leverage * Underlying Return

Effective leverage * Underlying Return

Path dependence of multi-day returns

Strong Medium(Only due to new lows or highs set by the underlying)

Mild (Only when stop-loss is breached)

Additional path variable (on which view is required)

Realized volatility of the underlying over the path

Worst (low/high) price of the underlying over the path

Whether the path breaches the stop-loss level

Core view required for multi-day bullish investment

“Underlying will move up but will do so in a trending/non-volatile manner”

“Underlying will move up but the minimum price in the interim will not be too low”

“Underlying will move up and the minimum price in the interim will not below the stop-loss price”

Dependence on issue/start date variables

No Yes Yes

Relative performance to static leverage

Outperforms(underperforms)

for large(small) moves in the underlying

Outperforms(underperforms) for adverse(favorable) moves in the underlying

Outperforms(underperforms) for adverse(favorable) moves in the underlying

Hedging frequency Daily When underlying makes new

min/max prices

When stop-loss level is breached


All three strategies are ways of controlling the risk of a leveraged position. They help ensure that (in absence of jumps) their future value does not become negative.

A major qualitative difference between LETFs and the other two strategies is that that the effective leverage embedded in the former is constant while that in latter changes with the value of the underlying.

For all three, the daily return of the underlying is simply equal to the effective leverage times the underlying return. However, the nature of multi-day returns is quite different. In all three cases, the multi-day return is not uniquely determined by the return of the underlying index and one needs to know additional information about the path leading up to the terminal value. However, the additional “path variables” required are quite different for each of these strategies. They are the realized volatility for the LETF, the worst price for self-funded futures and whether the worst price was lower than the stop-loss price for KO notes. We emphasize that all these path variables are related to volatility and increasing volatility is not beneficial.



Thus, in addition to a view on the underlying return, the investor is required to have a view on future value of these path variables. Which strategy is suitable depends on the particular view that the investor might have. The core view which would make a bullish investor prefer a long LETF is that he expects that the underlying will move up in a steady trending fashion. On the other hand, if he expects substantial inter-day volatility but does not expect the underlying to make substantial new lows he would probably prefer the self-funded futures or KO-note strategy.

In general, for a given terminal value of the underlying, these path variables have a distribution and this leads to path dependence. The corresponding expected payoffs for strategies for fixed terminal underlying prices are also quite different. Leveraged ETF underperform/outperform the static leveraged position for small/large moves in the underlying. On the other hand, KO notes outperform static leverage when terminal values are less than the stop loss price, underperform if terminal value is greater than stop loss but the stop loss has been hit and are in-line in all other cases. Self-funded futures are more complex and while they outperform for adverse moves they underperform for more favourable moves. We emphasize that the expected value for all three strategies across all possible terminal values of the underlying are of course the same and in this sense they are all “fair”. The difference is only that their values conditional on a given terminal value of the underlying are different.

Finally, we note that the amount of hedging that the manager/issuer of these instruments needs to do is quite different. As is well known, the LETF manager is required to rebalance his hedge on a daily basis. On the other hand, for self-funded futures, the rebalancing happens only when the underlying makes a new low (high) for a long (short) position. Finally, for the KO note, the hedge is unwound once the stop-loss barrier is breached.



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