How appropriate are leveraged ETFs for long-term investments?

Bachelor Thesis 2013 26 Pages

Business economics - Investment and Finance


Table of Contents

List of figures

List of tables

List of abbreviations

1. Introduction

2. Literature review
2.1. Compounding effect
2.2. Non-compounding effects
2.2.1 The effect of interest rates
2.2.2 The market impact of rebalancing
2.2.3 Counterparty risks and financial crises
2.2.4 Residual effects
2.2.5 Creation and redemption provision
2.3. Time horizon of investments

3. Analysis of LETFs in Germany
3.1. Overview
3.2. Objective
3.3. Methods

4. Results
4.1. Compounding effect
4.2. Fixed-percentage rebalancing and its frequency
4.3. Discussion of results

5. Summary and outlook


Numerical Sources

List of figures

Figure 1 Constructing a leveraged exchange-traded fund

Figure 2 Exemplified commitment to calculate average holding period

Figure 3 Rebalancing frequency

List of tables

Table 1 Two-day example for the compounding effect

Table 2 Overview of LETFs on the DAX

Table 3 Results from Monte-Carlo simulation

Table 4 Results after 5% rebalancing

List of abbreviations

illustration not visible in this excerpt

1 Introduction

Leveraged exchange-traded funds (LETFs) are actively managed funds that track the development of an underlying index with a factor Abbildung in dieser Leseprobe nicht enthalten= -1, 2 or -2[1] on a daily basis with the same amount of capital invested. A denomination of leveraged bull, bear and double leveraged bear ETFs is also used for the respective multiples of Abbildung in dieser Leseprobe nicht enthalten= 2, -1 and -2. Likewise, (double) inverse ETFs refer to the latter multiples as well.

A LETF manager can choose among several possibilities to achieve this factor. As depicted in Figure 1[2], he borrows additional funds to the amount of assets under management (AUM) from the financial markets and invests both, equity from the investor and debt, in an underlying index (buying on margin). Therefore, the fund manager usually has to pay interest on a daily basis, e.g. the London Interbank Offered Rate (LIBOR) or Euro OverNight Index Average (EONIA) as widely used reference interest rates. The importance of the interest payment will be discussed in section 2.2.

Abbildung in dieser Leseprobe nicht enthalten

More popular than the physical replication is the use of total return swaps (TRS) where a financial contract testifies that “one party (party A) makes a series of agreed payments and the other (party B) pays the total return on a particular asset.”[3] In our case the asset is a market index. Essentially, all major financial institutions offer TRS on common stocks, commodities, fixed income, foreign exchange and the main indices. Further, fund managers employ futures or other derivatives to achieve the promised ratio. A combination of the financial products mentioned is also possible.

In order to ensure a consistent leverage factor, the fund manager has to rebalance the portfolio on a daily basis[4] by raising or decreasing the exposure to the respective index to keep a constant percentage of leverage over time while the absolute amount can fluctuate.[5]

Since their inception in 2006, LETFs have been widely accepted because they serve several purposes in financial markets. For instance, Schubert (2011 and 2012) showed that inverse ETFs can effectively be used for hedging and focused on the ratio to be invested in the index fund and the inverse ETF respectively.[6] Additionally, on a long-term basis LETFs could be used for a tactical bet against a benchmark or to change the exposure to an index without physically altering the positions in the portfolio.[7] During financial crises and phases of low liquidity, inverse ETFs are useful substitutes for short-selling, especially when the underlying asset is hard to borrow.[8] Further, they do not require any financial literacy in short-selling and limit losses to the capital invested, in contrast to standard short positions.[9] Yet, Dobi and Avellaneda (2012) acknowledge that LETFs are hard to borrow due to arbitrage opportunities to capture the market impact of rebalancing explained below.[10]

In contrary, several multi-million dollar sentences in the United States in the last year[11] and growing concern about the long-term risks of LETF in the academic research has cast doubt on their reliability. Even DB X-Trackers, the ETF division of Deutsche Bank, state on their website that inverse ETFs are not appropriate for long-term investments.[12] Therefore, the central issue of this thesis is to clarify whether LETFs can effectively deliver the stated multiples for long periods. If they can and if markets rise in the long term, could investors simply buy and hold this asset class as a retirement provision and achieve their goal even two or three times faster? And if they cannot, what does a naïve investor need to take into account if he still wants to participate or has already engaged in such an investment? This thesis makes a contribution to the current status of research by being the first to explicitly analyze the German DAX and its respective leveraged funds.

It is organized as follows: Section 2 provides a discussion of compounding and non-compounding effects and a consideration of different time horizons. Section 3 analyzes the German LETF market and explains the objective and methodology employed to estimate the tracking error for different time horizons with and without rebalancing. Section 4 contains the results of the DAX analysis and refers back to section 2. The thesis concludes with a summary and an outlook in section 5.

2 Literature review

2.1 Compounding effect

Buying the German DAX index on the 13th of March 2003 for 2398.11 and selling it on the 23rd of April 2013 for 7658.21 would yield a return of 219% over 10 years.[13] Intuitively, a double LETF should yield approximately 438%. In reality, the LETF underperforms the underlying index and yields only 116%. More interestingly, a regular inverse ETF yields -64% instead of the expected -219%. This deviation is entirely caused by the compounding effect. The compounding effect occurs for holding periods greater than one day and is due to the daily fund rebalancing. Scientific literature has created the term “constant leverage trap”.[14] It causes an exponential compounding of gains and losses of previous periods and already after two days the cumulated return can vary substantially.

To illustrate the mathematical relationship, equation (1) describes the return of an underlying index after two days while equation (2) does the same for a leveraged bull ETF with a leverage factor of Abbildung in dieser Leseprobe nicht enthalten.[15] It is easy to show that the deviation equals twice the product of the individual period returns (equation (3)).

illustration not visible in this excerpt

The inverse leverage factorAbbildung in dieser Leseprobe nicht enthalteninduces even more deviation and yields a difference of six times the respectively daily returns:

illustration not visible in this excerpt

Interpreting this dependence concedes the intuition that a LETF yields a higher return when the underlying index successively moves into the same direction. In contrast, extreme volatility, in terms of changing signs, causes an underperformance of the LETF compared to its stated multiple.

To verify this assumption, Table 1 provides a numerical example by comparing the two-day performance of an underlying index with the respective leveraged bull and bear ETFs.[16] In cases 1 and 2, although the cumulative return corresponds, the LETF performance exceeds (case 1) or underperforms (case 2) the expected cumulated return. The same holds true for cases 3 and 4 and is in line with our expectations of changing signs. The last two cases carry the abnormality to the extreme. Although the index remains constant, an investor could loose a substantial part of his investment which can be attributed to the volatile return distribution of the example. This already indicates a general suspicion about the predictability of the long-term performance of a LETF.

illustration not visible in this excerpt

Table 1: Two-day example for the compounding effect, modified from Lu et al. (2009), p.19.

What generalizations can be drawn from these simple examples? In summary, the return deviation is path-dependent.[17] Several studies show that in periods of high volatility and modest returns the LETF underperforms the underlying index.[18] The reverse is true for periods of moderate volatility and a sufficiently large number of consecutive movements in either direction which is also documented by the simple two-day example presented above. Avellaneda and Zhang (2010b) established the term “time-decay”[19] and introduced a formula to define the volatility-contingent break-even point.[20] Although the path is critical for the magnitude of the deviation, the sequence does not matter, e.g. case 1 would yield the same results if the returns from the two periods were reversed.

Cheng and Madhavan (2009) showed that the rebalancing amount is dependent solely on the AUM of the LETF, its multiple and the underlying index return.[21] Based on this assumption, Tang and Xu (2011), derived a relationship between the compounding TE and the leverage factor, expressed with Abbildung in dieser Leseprobe nicht enthalten²-Abbildung in dieser Leseprobe nicht enthalten (Abbildung in dieser Leseprobe nicht enthaltenbeing the leverage factor).[22] Consequently, inverse leveraged funds deviate further than regular LETFs with the same absolute leverage ratio and LETFs with a multiple of -1 and 2 behave similarly.

2.2 Non-compounding effects

In contrast to the compounding effect, non-compounding effects are a major source for tracking errors particularly on a daily basis. In the following, the effect of interest rates, the market impact of rebalancing, counterparty risks and financial crises, the avoidance of excessive hedging costs, nonsynchronous trading, replication errors and the creation redemption provision will be discussed.

2.2.1 The effect of interest rates

Cheng and Madhavan (2009), Jarrow (2010) and Tang and Xu (2011) suggest that the LIBOR interest rate has a negative (positive) effect on the return deviation of leveraged bull (bear) ETFs[23] since fund managers mainly use derivatives such as total return swaps or forward contracts to deliver the promised multiple. Consequently, the interest payment (reception) impacts the daily performance. In general, all management-, hedging costs and other fees will be passed on to the fund. Contrariwise, I hypothesize that inverse ETFs should not be affected by this issue because the fund manager could retain the interest reception instead of passing it on to the fund.

Associated with the interest payments is the risk of changing interest rates. Within the last five years, the EONIA fluctuated as much as 4.6% in October 2008 to 0.056% in February 2013[24] and especially for long time horizons the interest rate cannot be estimated reliably. Thus, a LETF achieves the leverage factor only before fees and expenses. Yet, some funds set up lately explicitly promise to deliver the leverage factor even after fees and expenses.[25]

2.2.2 The market impact of rebalancing

Furthermore, Cheng and Madhavan (2009), Tang and Xu (2011) and Dobi and Avellaneda (2012) argue that the swap counterparty can predict the rebalancing behavior of the fund. At the end of each day, the fund manager has to increase (decrease) the exposure when the index value goes up (down) and thus rebalances in the same direction as the underlying index, independently of the leverage factor.[26] Cheng and Madhavan (2009) and Avellaneda and Zhang (2010b) denominated this management strategy as “buy high and sell low”[27]. Hence, the swap counterparty will offer the fund manager worse prices and the average tracking error should be even higher than the LIBOR or EONIA interest rate. Cheng and Madhavan (2009) derived a formula to quantitatively account for the deviation.[28] In contrast, Charupat and Miu (2011) argue that the swap counterparties themselves have outstanding positions from other liabilities which decrease the pricing arbitrariness.[29] Yet, one should not underestimate this effect, considering the large market impact of daily adjustment mentioned in section 2.1.

2.2.3 Counterparty risks and financial crises

Furthermore, counterparty risks are widely ignored when considering LETFs. Since most leverage is based on swap agreements, some researchers argue that if the counterparty becomes insolvent, it will not be able to meet its obligations and the fund may incur losses.[30] For example, when Lehman Brothers filed for bankruptcy on 15th of September 2008, it could not pay its obligations on time and ProShares, a major ETF issuer, had to cover possible losses.[31] In general, a financial crisis reduces market efficiency and liquidity and constrains credit conditions. It thus increases the cost of leverage and replication errors and limits the creation and redemption provision (both explained below). Consequently, investors should be aware that the deviation will be larger during the peak of the financial crises.

2.2.4 Residual effects

Moreover, nonsynchronous trading can explain the deviation of LETFs on internationally traded funds such as the MSCI Europe, Australia, and Far East or the MSCI Emerging Markets. When the respective LETFs are traded at the New York Stock Exchange (NYSE), the stock exchanges of emerging markets are closed and those of Europe are already finalizing their transactions. Thus, the leveraged funds incorporate supplementary information which may lead to mispricing and replication errors.[32] According to Shum (2010), all ETFs, whether leveraged or regular, are affected by this non-synchronization.[33]

Another effect introduced by Tang and Xu (2011) might be the avoidance of excessive hedging costs. A fund manager rebalances before the actual close of the stock market and will not observe the final value of the fund until the next day. However, as he always targets to minimize transaction costs, he is inclined to purchase fewer TRS and thus exhibits a preference for underexposure to overexposure to the index. If performed consistently, the tracking error will be negatively affected for leveraged bull ETFs and positively for leveraged bear ETFs.[34]

Finally, small errors can also be attributed to an incorrect replication of the underlying index itself. Schmidhammer et al. (2011) examined physical and synthetical replication strategies of the German equity index DAX and found discrepancies in the pricing efficiency.[35] Elia (2012) did not analyze exclusively the German DAX but noted similar results for different European ETFs.[36] Hence, the offered LETF may already be traced incorrectly in the first place.

2.2.5 Creation and redemption provision

In contrast, the creation and redemption provision is one force that minimizes tracking errors. Towards the end of each trading day, the net asset value (NAV) is disclosed to a small number of authorized participants that use this arbitrage mechanism to drive the market price back to its NAV.[37] The pricing efficiency is even stronger for leveraged ETFs (compared to traditional ETFs) since the creation and redemption is executed in cash rather than in kind. This can be attributed to the use of derivatives to replicate the leverage instead of holding a basket of securities which reduces transaction costs and thus facilitates arbitrage transactions. The exact arbitrage boundaries depend on the transaction costs and the construction of the provision.[38] Yet, high volatility during the last hours of trading might reduce the effectiveness of the creation and redemption provision.[39]

In general, Shum and Kang (2012) calculated correlations between compounding and non-compounding effects and argue that they do not reinforce each other. Contrariwise, because both effects can occur in either direction, the LETF might even experience a TE reduction through opposing deviations.[40] In addition, several studies suggest that the non-compounding effects have a higher impact than the simple compounding effect.[41]

2.3 Time horizon of investments

Understanding the reasons for a deviation from the stated multiple, it remains essential for an investor to know when the tracking error becomes significant. Lu et al. (2009) assume that for double and double inverse ETFs a holding period of one month or less is within a safe scope for the investor[42]. Charupat and Miu (2011) are stricter by showing that the performance diverges for holding periods longer than a week (e.g. a month).[43] Tang and Xu (2011) estimate that large deviations occur after multiple holding days.[44] Shum and Kang (2012), who focused mainly on non-compounding effects, object that the error is even higher on a one-day level than after one week or one month.[45] Lu et al. (2009) discovered a similar phenomenon for US LETFs[46] but Charupat and Miu (2011) assume that this is can be explained by the use of different derivatives to track the returns. US LETFs are replicated through exchange-traded futures, while the Canadian LETFs analyzed by Charupat and Miu (2011) are generated through TRS and forwards.[47] Contrarily, Hill and Foster (2009), two fund managers, show that LETFs deliver the stated multiple reliably even for longer holding periods.[48] However, comparing the beta ranges of the different studies, it becomes apparent that the latter interpret the ranges much more progressively.

Schubert (2010) and Funke et al. (2012) exclusively analyzed the average holding period of LETFs of individual investors in the German market.[49] Their results seem to suggest that German investors are aware of tracking errors and thus limit the holding period. Yet, there is a major drawback to their analysis: Both papers cite Figure 2, explaining that the average holding period of commitments with several purchases and/or disposals is calculated by dividing the integral by the sum of all purchases. In the example below, the average holding period is Abbildung in dieser Leseprobe nicht enthaltendays which is 71.32% of the total holding period (19 days).

Abbildung in dieser Leseprobe nicht enthalten

Figure 2: Exemplified commitment to calculate average holding period, mod. from Schubert (2010), p.12.

Why is this calculation misleading? Hypothetically, investor A purchases 100 shares, holds them for 50 days and subsequently sells them. Likewise, investor B purchases 100 shares at the same time but buys additional 400 shares on day 49 and sells all 500 shares on day 50. Both investors hold essentially the same amount for the same time. However, in the first case the average holding period equals Abbildung in dieser Leseprobe nicht enthaltendays whereas the latter is Abbildung in dieser Leseprobe nicht enthaltendays which is only 22% of the total holding period.

Thus, on average, the holding period is shorter since this effect is one-sided, e.g. it is not possible to increase the average by adjusting only the purchase or disposal amount. On the one hand, only 10% of the commitments are impacted by this mistreatment. On the other hand, the average holding period is used for further analysis and hence affects the average trade volume and other positions. Therefore, their results will not be considered further. As both studies have been sponsored by Deutsche Bank AG and Comdirect Bank AG, it is valid to assume that this downward bias is in the interest to document an appropriate handling of LETFs by private investors.

In general, all studies consistently point out that inverse ETFs have a larger tracking error as the time horizon widens, compared to their bull counterparts. The calculations conducted in this work have generated similar results. As the holding period increases, the relationship between the ETF and the index is disrupted. This effect becomes significant at the one-month level and is stronger for leveraged bear ETFs. The results will be explained in further detail in section 4 of this thesis.

3 Analysis of LETFs in Germany

Keeping in mind the criticism mentioned in section 2.3, further insight into the performance of LETFs is needed. This paper contributes to the current research by taking a closer look at the German LETF market. The German DAX has been chosen because it is an economically important, actively traded and representative equity index that has hardly been taken into consideration by academic research. The following section provides an overview of the German DAX and LETFs that use it as their underlying index.

3.1 The German DAX

The DAX (Deutscher Aktienindex) is the most important German stock market index and comprises 30 leading and top-selling stocks of German companies at the Frankfurt stock exchange. The weighting is based on the volume of stocks traded, market capitalization (based on free float) as well as industry representativeness using the Lasypeyres index formula and it is calculated every second. It was first issued on 30th of December 1987 and until 2009 Deutsche Börse AG held the trademark rights. After a sentence of the German BGH on the 30th of April 2009, the DAX has become common property.[50] The DAX is a performance or total return index (in opposite to a price index) and thus it is calculated as if all dividend payments and subscription rights were reinvested. The DAX exists also as a price index which is insignificant to the financial public.

The guidelines of Deutsche Börse AG (2011) explain how the leverage indices are calculated and incorporate a safety mechanism to protect investors from abnormal daily losses.[51] When the underlying index declines more than 25% in one day, the loss of the LETF is restricted to 50%. However, this never occurred for the German DAX. Additionally, if the underlying index gains or looses more than 40% in a month, the indices will be readjusted as well.

At the Frankfurt stock exchange, currently more than 1.000 equity ETFs are listed that track the performance of the equity market of a country, a region, a sector or a strategy, of bonds, commodities or currencies. Furthermore 25 inverse ETFs, 15 double ETFs and 6 double inverse ETFs are available for the respective equity markets.[52] The leveraged ETFs tracking the DAX manage a total volume of nearly one billion € and are listed below:

illustration not visible in this excerpt

Table 2: Overview of LETFs on the DAX, volume as of 30th of April 2013 in €m.

The first LETF on the DAX, Lyxor ETF LevDAX, was issued on the 29th of June 2006 and thus earlier than the first US LETF offered by Rydex in late 2006[53]. LETFs demand a total expense ration (TER) in the range of 0.3% to 0.6% which account for a small fraction of the tracking error. Compared to other funds, the expenses are higher than those of passively managed funds such as db x-trackers DAX® UCITS ETF or Lyxor ETF DAX, both with a TER of 0.15%, but lower than the TER of approximately 1%-2% of actively managed investment funds.

Interestingly, Deutsche Bank AG offers an ETF simulator[54] where private and institutional investors can vary the leverage factor, assumptions on volatility and return as well as the time horizon of the simulation to estimate the development of a LETF especially considering the path dependency. After a simulation of two days, the leveraged funds never meet the stated multiple. Simulating 1.000 days with a high volatility shows the extreme deviations that can occur over a long holding period. Nevertheless, such tools contribute to a better comprehension of the embedded risks and benefits of LETFs among institutional and particularly (naïve) private investors.


[1] US funds offer leverage factors of (-)3 and (-)4 as well.

[2] Modified from Avellaneda and Zhang (2010), p.54.

[3] Brealey et al. (2011), p.691.

[4] Deutsche Börse AG also offers ETFs with a monthly rebalancing which will not be considered further in this thesis.

[5] See Hill and Teller (2009), p.69 or Dobi and Avellaneda (2012), p.5f. for a practical guidance on how to rebalance as a fund manager.

[6] See Schubert (2011), p.24 and Schubert (2012), p.76.

[7] See Hill and Foster (2009), p.4.

[8] See Avellaneda and Zhang (2010a), p.54.

[9] See Shum and Kang (2012), p.2.

[10] See Dobi and Avellaneda (2012), p.15.

[11] See Bloomberg (2012), Reuters (2012) and Financial Industry Regulatory Authority (2012).

[12] See Deutsche Bank (2013a).

[13] Own calculations based on data from Comdirect (2013).

[14] Trainor and Baryla (2008), p.49.

[15] See Lu et al. (2009), p.4f.

[16] The daily returns are invented to illustrate tracking errors. Of course, the German DAX never exhibited such large movements in two consecutive days.

[17] See also Jarrow (2010), p.137.

[18] See Lu et al. (2009), p.7 or Avellaneda and Zhang (2010b), p.587.

[19] Avellaneda and Zhang (2010b), p.594.

[20] See Avellaneda and Zhang (2010b), p.597.

[21] See Cheng and Madhavan (2009), p.47.

[22] See Tang and Xu (2011), p.13.

[23] See Cheng and Madhavan (2009), p.57, Jarrow (2010), p.137 and Tang and Xu (2011), p.8.

[24] See Global Rates (2013).

[25] See Cheng and Madhavan (2009), p.45.

[26] See Cheng and Madhavan (2009), p.48f., Tang and Xu (2011), p.9f. and Dobi and Avellaneda (2012), p. 6f.

[27] Cheng and Madhavan (2009), p.57 and Avellaneda and Zhang (2010b), p.589.

[28] See Cheng and Madhavan (2009), p.58.

[29] See Charupat and Miu (2011), p.976.

[30] See Cheng and Madhavan (2009), p. 60, Elia (2012), p.3 and Shum and Kang (2012), p.4.

[31] See Loehr and Lamb (2013), p.29.

[32] See Charupat and Miu (2011), p.973 and Shum and Kang (2012), p.12.

[33] See Shum (2010), p.3.

[34] See Tang and Xu (2011), p.10.

[35] See Schmidhammer et al. (2011), p.348.

[36] See Elia (2012), p.14.

[37] See Shum and Kang (2012), p.2.

[38] See Charupat and Miu (2011), p.968.

[39] See Tang and Xu (2011), p.11.

[40] See Shum and Kang (2012), p.5.

[41] See Tang and Xu (2011), p.5 and Dobi and Avellaneda (2012).

[42] See Lu et al. (2009), p.15.

[43] See Charupat and Miu (2011), p.975.

[44] See Tang and Xu (2011), p.28.

[45] See Shum and Kang (2012), p.11.

[46] See Lu et al. (2009), p.22.

[47] See Charupat and Miu (2011), p.975.

[48] See Hill and Foster (2009), p.3 and p.14.

[49] See Schubert (2010), p.11 and Funke et al. (2012), p.13f.

[50] See Handelsblatt Online (2009).

[51] See Deutsche Börse (2011), p.25ff.

[52] See Deutsche Börse (2013).

[53] See Avellaneda and Zhang (2010b), p.586.

[54] See Deutsche Bank (2013b).


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ETF Investment strategy LETF DAX




Title: How appropriate are leveraged ETFs for long-term investments?