Alpha Dark Horse or Trojan Horse? A Comprehensive Analysis of Leverage Financial Products

1. Introduction

Despite warnings from many studies (Trainer and Baryla, 2008; Humphries, 2011; Lu et al., 2012; Pessina and Whaley, 2021), investor demands for leveraged products seem insatiable as more and more new products related to single stocks are coming to the market1. Most investors, however, are unaware of the risks associated with leveraged products, mistaken them as tools for producing excess returns. As the asset under management (AUM) and number of funds available increases, so are the losses suffered by investors when market volatility increases. Johnson (2025) reported that investors lost more than $25 billion on leveraged ETFs in one day when the US announced a tariff war in April 8th, 2025.

Two noteworthy products can illustrate the complexity and potential harm to investment return of these products: the TSL3.L (3X Long Tesla Shares) and SOXL (3X Long SOXX ETF). Between the height of Tesla’s stock price in December 17th, 2024 and March 17th, 2025, the TSL3.L dropped from $116 to $8.7, a decline of 93%, where the underlying stock price had declined by 50% during that period (from $480 to $240). The SOXX had a six-month high of $237 in January 22, 2025 and low of $155 in April 8th, 2025 (a 35% decline). The 3X product SOXL, however, saw the price declined from $35 to $8.5 (decline of 76%). When Tesla’s stock price recovered from $240 in April 8th to $300 in June 8th, the TSL3.L only gone up to $9.7 (an 11.5% increase compared to 25% increase in the underlying stock’s price). SOXX went from $155 in April 8th, 2025 to $222 in June 9th, 2025 (a 43% increase), while the 3X product went from $8.5 to $20.6 (138% increase, more than 3X of 43% increases in the underlying ETF). From October 10th, 2022 to July 8th, 2024, the price of SOXX increased from $98 to $260 (165.31% increase). The price of SOXL went from $6.5 to $64, or 884.62% increases, 5.3X the return for the underlying asset’s price increases. It is clear that there are important factors that drive the deviations from their stated leverages for single stock products and index products.

The main driving forces for leveraged products return to deviate from their underlying assets (decay) are leverage ratio, time, return of the underlying asset and volatility (Trainor and Caroll, 2013). The vastly different return patterns for the two products listed above suggest that there are other factors at play. This paper aims to provide comprehensive analysis of leverage products and identify the risks that these products could bring to investors, regulators, and the financial market a whole. Our findings suggest that the nature of the underlying assets plays the most critical role in determining the risk and return of a leveraged product. The volatility of the underlying assets, which is the main driving force for investor interests, is the second most critical factor. A third factor, which is not discussed frequently in existing literature, is the fund flow and trading volume of the product. Because of the need for daily rest for most of these products, an unchecked growth in these products could distort the price discovery function of the derivative products used by these leveraged products, and introduce significant systemic risks due to increasing reliance on over-the-counter swaps and derivatives. Based on our findings, we also provide policy suggestions for investors, fund managers, and policymakers.

The remainder of this paper is structured as follows: Section 2 provides a comprehensive review of academic research on leveraged ETFs, including studies on performance characteristics, volatility effects, market impact, and investor behavior. We also review literature on the risks associated with the futures and options markets that underpin leveraged ETF operations. Section 3 outlines our approach to measuring leveraged ETF performance, including the calculation of tracking errors, volatility drag, and risk-adjusted returns. This section also presents theoretical models of leveraged ETF decay and describes our simulation methodology for analyzing the effects of different volatility regimes on leveraged ETF performance. Section 4 presents our empirical findings based on five years of daily data for a representative sample of leveraged ETFs across different asset categories. We analyze performance patterns, quantify tracking errors and volatility drag, and examine how these metrics vary across different market conditions and asset classes. Section 5 summarizes our key findings. We conclude the paper in section 6 by discussing the implications of our findings for investors, financial advisors, and regulators, and suggests directions for future research.

2. History of Leveraged Products

The first leveraged ETF was introduced by ProFunds Group in 2006, with their ProShares Ultra S&P500 (SSO) offering 2x daily exposure to S&P 500 returns. This innovation was quickly followed by inverse and 3x leveraged products, creating an entirely new category of exchange-traded products. Unlike conventional ETFs that aim to track the performance of an underlying index on a one-to-one basis, leveraged ETFs seek to deliver multiples (typically 2x or 3x, with an extreme case of 5x Mag 7 stocks) of the daily returns of their benchmark indices. Inverse leveraged ETFs aim to deliver multiples of the opposite performance of their benchmarks. These products achieve their leverage through the use of derivative instruments such as futures contracts, swaps, and options, combined with debt financing.

The market for leveraged ETFs has grown substantially since their inception. As of 2025, there are over 250 leveraged and inverse ETFs trading in U.S. markets, with combined assets under management exceeding $100 billion. The largest issuers include ProShares, Direxion, and UBS, offering products across virtually every asset class including domestic and international equities, fixed income, commodities, currencies, and volatility indices. The most popular leveraged ETFs by trading volume are those tracking major equity indices such as the S&P 500, NASDAQ-100, and Russell 2000, with the technology and financial sectors also attracting significant investor interest.

The growth of leveraged ETFs represents a significant development in financial markets, both in terms of assets under management and daily trading volume. While leveraged ETFs account for approximately 2% of the total ETF market by assets, they often represent a disproportionately large share of daily ETF trading volume, sometimes exceeding 20% of all ETF trades on high-volatility days. This outsized trading activity reflects their popularity among active traders and short-term investors who use these instruments for tactical asset allocation, hedging, and speculative purposes. As Chan and Lien (2006) point out, speculators tend to migrate to more risky derivative products and could render the price discovery function of these products useless. Therefore, leverage products could make induce more volatilities into the market and could potentially create higher systemic risk.

The significance of leveraged ETFs extends beyond their direct asset base. These products have become important components of market microstructure, particularly during periods of market stress. The daily rebalancing requirements of leveraged ETFs can generate substantial trading volume near market close, potentially amplifying market movements and contributing to end-of-day volatility. Shum and Kang (2013) suggests that the hedging activities of leveraged ETF issuers may have contributed to market dislocations during periods of extreme volatility, such as during the 2008 Financial Crisis and March 2020 COVID-19 market crash.

From a regulatory perspective, leveraged ETFs have attracted significant attention from bodies such as the Securities and Exchange Commission (SEC) and the Financial Industry Regulatory Authority (FINRA). Both organizations have issued investor alerts and guidance regarding the risks associated with these products, particularly emphasizing their unsuitability for buy-and-hold investors. In 2020, the SEC adopted new rules requiring broker-dealers to perform enhanced due diligence before approving retail investor accounts for leveraged ETF trading, reflecting ongoing regulatory concerns about investor understanding of these complex products.

2.1. Types of Leverage Products and Their Structures

Leveraged ETFs can be categorized along several dimensions, including the direction of leverage, the magnitude of leverage, the underlying asset class, and the legal structure of the product.

Direction of Leverage: - Bull (Long) Leveraged ETFs: These aim to deliver positive multiples (e.g., 2x or 3x) of the daily returns of their underlying indices. Examples include ProShares UltraPro S&P500 (UPRO) and Direxion Daily Semiconductor Bull 3X Shares (SOXL). - Bear (Short/Inverse) Leveraged ETFs: These aim to deliver negative multiples (e.g., -1x, -2x, or -3x) of the daily returns of their underlying indices. Examples include ProShares UltraPro Short QQQ (SQQQ) and Direxion Daily Financial Bear 3X Shares (FAZ).

Magnitude of Leverage: - Standard Leverage (2x): These products aim to deliver twice the daily return of their underlying indices. - Enhanced Leverage (3x): These products aim to deliver three times the daily return of their underlying indices. - Ultra Leverage (4x): A small number of products offer even higher leverage factors (5x Mag 7 stocks, for example), though these remain relatively rare due to regulatory concerns.

Underlying Asset Classes: - Equity Indices: The most common type, tracking broad market indices (S&P 500, NASDAQ-100), sector indices (financials, energy, technology), or international markets. - Fixed Income: Tracking various bond indices, including Treasuries, corporate bonds, and high-yield bonds. - Commodities: Providing leveraged exposure to commodities such as gold, oil, and agricultural products. - Currencies: Offering leveraged exposure to currency pairs or baskets. - Volatility: Tracking volatility indices such as the VIX.

2.2. Mechanism of Leverages

Leveraged ETFs (LETFs) can achieve their desire leverage by using a combination of these 4 methods:

• Borrowing funds and buy the underlying assets on margin.
• Using futures contracts.
• Using Options contracts.
• Using Swaps.

Method 1 is relatively straightforward and carries the least amount of tracking errors and volatility drag as the product would not require daily resetting. The issue with this method is an investor could easily carry out on their own (if similar margin requirements applied for these products). The sole drag on return would be the cost of margin. Unless there is substantial spread between the cost of borrowing for the leveraged product sponsor and retail investor, the benefit of these type of LETFs is not clear.

Method 2 is the most problematic, especially for the LETFs that are only based on futures contracts (as is true for most commodity-based LETFs, see Pessina and Whaley, 2021 for a more detailed analysis). Futures contracts are leveraged products already. Most futures contracts have initial margins that would result in more 10x leverage. Take the WTI Crude Oil Futures (the main tool for ProShare Ultra Bloomberg Crude Oil (UCO)) as an example. It has a contract specification of 1,000 barrels of WTI, with initial margin of around $5,500 per contract for the near month contracts. With the price of WTI at $64.77 per barrel in June 10^th, 2025, it represents a 11.78x leverage. To achieve 2x leverage, the fund sponsor only have to invest less than 20% of the fund in the futures contracts. The tracking error for these type of products are notorious high (on the negative side) due to the potential for front running when these funds roll over their futures position (Chan et al., 2020). Since the inception in December, 2008 (when the price of WTI was about $40 a barrel), the return for UCO is -99.6%. The performances for leveraged commodity products were so bad that three 3x WTI products DWTI, UWT, and OILU) were delisted due to prices falling too low. The most recent offering 3CRL will likely face the same fate as well.

The use of options is likely the more complicated strategy and could result in the highest amount of tracking errors. The straightforward method would be to use deep-in-the-money calls to produce the desire leverage. For example, in June 10^th, 2025 the price of QQQ gone up by $3.51, or 0.66% ($530.7 to $534.21). To achieve the 3x leverage outcome, the option premium that would give the approximately 2% change in price would be the options with premium of around $175 in the previous day. This would require the sponsor for TQQQ to buy options with strike price of $350 that are expiring within a few days. There is no evidence in the trading volume to suggest anyone is doing that. However, if the sponsor purchased near-the-money calls with relatively short maturity and roll the contracts over periodically, in an up cycle, the TQQQ performance would be better than the 3x stated leverage ratio. On the other hand, if the market turns negative drastically, the TQQQ with mostly options and cash could see negative return more than -3x the underlying asset. This strategy, while produces the highest degree of tracking errors, also protects the investor as the cash holdings cushions the downside risks. Our findings indicate that this is likely the reason behind the deviation experienced by SOXL.

The use of swaps is the strategy that will result in almost no tracking error in theory. For example, a 2x long fund could enter into a swap agreement with a -2x short fund to buy the underlying asset at the previous closing price to be delivered at the end of today’s closing. Suppose that the closing price today represents a 1% gain, then the long position would gain 2%, while the short position would lose 2%. In a perfect world, there will be sufficient demands for both long and short positions for swaps to meet all the needs of the leveraged products. But the chance of a perfect match is close to zero. As a result, the sponsor would be required to enter into swap agreements with OTC counter parties, and/or use additional strategies to try to mimic the 2x outcome.

With the exception of strategy 1, all other strategies would result in losses from increases in bid-ask spread, especially if the fund is large enough to necessitate brokers/dealers to act as market makers. The increases in demand for swaps, futures, and options from leverage products could contribute to much higher systemic risks when the market for these products gets to be big enough.

2.3. Potential Risks to Financial Markets and Investors

Leveraged ETFs present several distinct categories of risk that differentiate them from traditional investment vehicles. These risks operate at both the individual investor level and the broader systemic level.

Individual Investor Risks:

• Compounding and Volatility Decay: Perhaps the most significant risk is the mathematical reality that leveraged ETFs are designed to achieve their stated multiple of returns on a daily basis only. Over longer periods, the effects of compounding can cause returns to deviate significantly from the intuitive expectation of the leverage multiple times the underlying index return. This deviation, known as "volatility drag" or "beta slippage," typically increases with both the holding period and the volatility of the underlying index. Our empirical analysis shows that for a 3x leveraged ETF like TQQQ, the volatility drag can exceed 14x over a five-year period.
• Path Dependency: The returns of leveraged ETFs are highly path-dependent, meaning that two different price paths leading to the same endpoint for the underlying index can result in dramatically different leveraged ETF returns. This makes performance highly unpredictable over longer time horizons.
• Complexity and Misunderstanding: Many retail investors fail to fully understand the daily reset mechanism and its implications for longer-term performance. Survey data suggests that a significant percentage of leveraged ETF holders incorrectly believe these products will deliver their stated leverage multiple over periods longer than one day.
• Execution and Liquidity Risks: While the largest leveraged ETFs maintain adequate liquidity, smaller or more specialized products can experience wide bid- ask spreads, particularly during market stress. This can increase transaction costs and execution risk.
• Counterparty and Credit Risks: Leveraged ETFs rely heavily on derivatives such as swaps and futures contracts, introducing counterparty risk. For leveraged ETNs, investors also face the credit risk of the issuing financial institution.

Systemic Risks:

• End-of-Day Rebalancing Effects: The need for leveraged ETF issuers to rebalance their portfolios near market close to maintain their target leverage ratios can create predictable trading patterns that may be exploited by other market participants or potentially amplify market movements.
• Derivatives Market Impact: The substantial use of derivatives by leveraged ETF issuers can impact pricing in futures and swaps markets, potentially affecting other market participants who use these instruments for hedging or investment purposes.
• Volatility Amplification: During periods of market stress, the rebalancing activities of leveraged ETFs can potentially amplify market volatility, creating a feedback loop where increased volatility necessitates larger rebalancing trades, which in turn may further increase volatility.
• Concentration Risk: The popularity of certain leveraged ETFs tracking specific sectors or indices can lead to concentration of derivative positions, potentially creating systemic vulnerabilities if multiple issuers need to execute similar trades simultaneously during market disruptions.
• Regulatory Concerns: The complexity and potential for misuse of leveraged ETFs has attracted regulatory scrutiny, with concerns about their suitability for retail investors and potential contribution to market instability.

Our research indicates that these risks are not merely theoretical. The empirical data analyzed in this paper demonstrates significant performance divergence between leveraged ETFs and their underlying indices over multi-year periods, with particularly pronounced effects in high-volatility asset classes such as technology and semiconductor sectors.

2.4. Research Objectives and Paper Structure

This paper aims to provide a comprehensive analysis of leveraged ETFs/ETNs, with particular focus on quantifying their risks, understanding their performance characteristics across different market conditions, and evaluating their potential impact on financial markets. Specifically, our research objectives are:

• To analyze the historical performance of leveraged ETFs across different asset categories, leverage factors, and time horizons, with particular attention to the divergence between actual returns and the theoretical multiple of underlying index returns.
• To quantify the volatility drag and tracking errors experienced by leveraged ETFs under different market conditions (bull vs. bear markets, high vs. low volatility environments).
• To examine the relationship between underlying asset volatility and leveraged ETF performance decay, testing the theoretical prediction that higher volatility leads to greater decay effects.
• To evaluate the risk-return characteristics of leveraged ETFs compared to their underlying indices and alternative investment strategies.
• To assess the potential systemic risks posed by leveraged ETFs, particularly with respect to their daily rebalancing requirements and use of derivatives.

2.5. Literature Review

The literature on leveraged ETFs has grown substantially since these products were introduced in the mid-2000s. This section provides a comprehensive review of the key research, focusing on performance characteristics, volatility effects, market impact, and investor behavior.

2.5.1. Performance Characteristics and Path Dependency

One of the most extensively studied aspects of leveraged ETFs is their performance over holding periods longer than one day. Avellaneda and Zhang (2010) provided one of the first rigorous mathematical analyses of leveraged ETF returns, demonstrating that the performance of these products is path-dependent and generally underperforms the leveraged multiple of the underlying index over longer periods. Their stochastic calculus model showed that the expected return of a leveraged ETF over a multi-day period is approximately equal to the leveraged multiple of the index return minus a decay term proportional to the variance of the index and the square of the leverage ratio.

Building on this theoretical foundation, Cheng and Madhavan (2009) examined the impact of daily rebalancing on leveraged ETF performance. They demonstrated that in volatile but trendless markets, both bull and bear leveraged ETFs can lose value, creating a "volatility decay" effect. Their analysis showed that for a 2x leveraged ETF, if the underlying index experiences daily returns of +10% followed by -10%, the leveraged ETF would lose approximately 4% over the two-day period, despite the index ending at roughly the same level.

Tang and Xu (2013) extended this analysis by examining the performance of leveraged ETFs during the 2008-2009 financial crisis, a period of extreme market volatility. They found that leveraged ETFs significantly underperformed their stated multiples during this period, with some 3x leveraged ETFs delivering less than 2x the underlying index return over monthly periods. This underperformance was particularly pronounced for leveraged ETFs tracking more volatile indices. Volatility is a foe rather than a friend for leveraged product investors. However, investors who are interested in these products are likely drawn to them because of the higher volatility, particularly when the prices of these products have gone up significantly recently.

In a comprehensive empirical study, Hill and Foster (2009) analyzed the returns of leveraged ETFs across different market conditions and holding periods. They found that the tracking error between leveraged ETF returns and the leveraged multiple of index returns increased with both the holding period length and the volatility of the underlying index. For holding periods of one month or longer, the tracking error became substantial enough to significantly impact investment outcomes.

Curcio and Dickerson (2017) examined the long-term performance of leveraged ETFs over an eight-year period and found that, contrary to the conventional wisdom that leveraged ETFs are unsuitable for long-term holding, some bull leveraged ETFs delivered exceptional returns during sustained market uptrends, outperforming even their theoretical perfect leverage multiple. However, they noted that this outperformance was highly dependent on the specific market conditions and unlikely to be replicated in more volatile or downward-trending markets. We saw evidence of such characteristics for the returns for SOXL v.s. SOXX from Covid-19 period to mid-2024, as discussed in the introductory section of this paper.

2.5.2. Volatility Effects and Risk Characteristics

The relationship between market volatility and leveraged ETF performance has been another major focus of academic research. Leung and Ward (2015) developed a mathematical model linking the volatility of the underlying index to the expected decay in leveraged ETF returns. Their model demonstrated that the expected decay is proportional to the square of the volatility and the square of the leverage ratio minus the leverage ratio itself. This implies that 3x leveraged ETFs experience approximately nine times the volatility decay of their underlying indices. Dobi and Avellaneda (2012) introduced the concept of "volatility threshold" for leveraged ETFs, defining it as the level of volatility at which the expected return of a leveraged ETF becomes negative despite a positive trend in the underlying index. They calculated that for a 3x leveraged ETF, this threshold is reached when the annualized volatility exceeds twice the annualized drift of the underlying index.

In terms of risk characteristics, Charupat and Miu (2011) analyzed the volatility and beta of leveraged ETFs relative to their underlying indices. They found that while daily volatility and beta closely matched the theoretical expectations (i.e., 2x or 3x the underlying index), these relationships broke down over longer measurement periods. Over monthly periods, the realized beta of leveraged ETFs was often significantly lower than their stated multiple, particularly during high volatility regimes. Guedj et al (2011) examined the risk-adjusted performance of leveraged ETFs using various metrics including Sharpe ratio, Sortino ratio, and maximum drawdown. They found that on a risk-adjusted basis, leveraged ETFs generally underperformed both their underlying indices and unleveraged ETFs tracking the same indices. This underperformance was attributed to the combined effects of volatility decay, management fees, and transaction costs associated with daily rebalancing.

Loviscek, Tang, and Xu (2014) investigated the diversification properties of leveraged ETFs in portfolio contexts. Contrary to conventional portfolio theory, they found that including inverse leveraged ETFs in a portfolio did not necessarily provide effective diversification benefits during market downturns due to the volatility decay effect. They concluded that the path dependency of leveraged ETF returns makes them unreliable hedging instruments over multi-day periods.

2.5.3. Market Impact and Systemic Risk

A growing body of literature has examined the potential impact of leveraged ETFs on broader market dynamics, particularly concerning end-of-day trading patterns and potential systemic risks. Tuzun (2014) analyzed the rebalancing activities of leveraged ETFs and their impact on market volatility. Using a dataset of leveraged ETF holdings and trading activities, he estimated that leveraged ETF rebalancing could account for approximately 16-22% of daily trading volume in the last hour of trading during periods of high market volatility. Building on the work of Tuzun (2014), Ivanov and Lenkey (2018) examined the price impact of leveraged ETF rebalancing on underlying securities. They found evidence of temporary price pressure effects, particularly for less liquid stocks included in the indices tracked by leveraged ETFs. Their analysis suggested that these price pressures could create profitable trading opportunities for sophisticated market participants who anticipate leveraged ETF rebalancing flows.

Ben-David et al (2020) and Bhattacharya and O’Hara (2017) investigated whether the growth of ETFs and other synthetic investment products had contributed to increased market risk and fragility. They found some evidence of amplified volatility during extreme market movements, reducing the informational efficiency of the underlying asset’s markets. LETFs will likely amplify these inefficiencies. Shum et al. (2016) presented additional evidence, suggesting that leveraged ETFs could contribute to market instability through feedback effects. They developed a model showing how the mechanical rebalancing requirements of leveraged ETFs could create a positive feedback loop during market stress: increased volatility necessitates larger rebalancing trades, which in turn may further increase volatility. Their empirical analysis of the 2015 market turbulence provided some support for this theoretical mechanism.

2.5.4. Investor Behavior and Product Usage

Research on how investors actually use leveraged ETFs has provided important insights into potential mismatches between product design and usage patterns. Cheng and Madhavan (2009) were among the first to raise concerns about retail investors potentially misunderstanding the daily reset feature of leveraged ETFs and inappropriately holding these products for extended periods.

Empirical evidence for these concerns was provided by Guedj et al. (2011), who analyzed account-level data from a major brokerage firm. They found that retail investors held leveraged ETFs for an average of 15 trading days, significantly longer than the one-day horizon for which these products are designed. Moreover, they documented that investors who held leveraged ETFs for longer periods experienced worse performance outcomes on average.

In a survey-based study, Bhattacharya et al (2017) found that a significant percentage of retail leveraged ETF investors did not fully understand the daily reset mechanism and its implications for longer-term performance. Approximately 30% of surveyed investors incorrectly believed that a 2x leveraged ETF would deliver twice the return of the underlying index over a one-month period. Giese (2010) investigated the use of leveraged ETFs in portfolio construction. He demonstrated that while conventional wisdom suggests leveraged ETFs are unsuitable for buy-and-hold strategies, there are specific market conditions under which strategic allocations to leveraged ETFs could enhance portfolio performance. However, he emphasized that such strategies require sophisticated risk management and continuous monitoring.

2.6. Research on Futures and Options Risks

Since leveraged ETFs achieve their leverage through derivatives such as swaps, futures and options, understanding the risks associated with these instruments is crucial for comprehending the overall risk profile of leveraged ETFs and the potential effects they have on the financial system.

2.6.1. Basis Risk and Roll Yield

Futures-based leveraged ETFs (such as UCO) are subject to basis risk and roll yield effects, which have been extensively studied in the academic literature. Erb and Harvey (2006) provided a seminal analysis of the "roll yield" in commodity futures markets, demonstrating how the shape of the futures curve (contango vs. backwardation) can significantly impact returns for investors in futures-based products. They showed that in contango markets, where futures prices are higher than spot prices, the process of "rolling" futures contracts forward results in negative returns even if the underlying spot price remains unchanged. Building on this work, Mou (2011) examined the impact of roll yield on leveraged commodity ETFs. He found that the negative roll yield in contango markets was amplified by leverage, creating an additional source of decay beyond the volatility effect. For a 2x leveraged ETF tracking a commodity index in a steep contango market, the roll yield could reduce returns by more than 1% per month. Aulerich et al (2013) investigated the market impact of futures rolling by index funds and ETFs. They found evidence that the predictable rolling patterns of these funds created profitable front-running opportunities for other market participants, potentially increasing the costs borne by ETF investors. This effect was particularly pronounced for leveraged ETFs due to their larger and more frequent rolling requirements.

2.6.2. Counterparty and Liquidity Risks

Leveraged ETFs that use swaps and other over-the-counter derivatives face counterparty risks that have been analyzed in several studies. Cont and Kokholm (2014) developed models for quantifying counterparty risk in swap agreements, showing how this risk increases during periods of market stress when the probability of counterparty default rises precisely when the swap value is highest. Liquidity risks in derivatives markets were examined by Brunnermeier and Pedersen (2009), who developed the concept of "liquidity spirals" where funding liquidity and market liquidity can mutually reinforce each other during crisis periods. Their model has implications for leveraged ETFs that rely on liquid derivatives markets to maintain their leverage ratios. Empirical evidence of these liquidity risks was provided by Bollen, O'Neill, and Whaley (2017), who analyzed the liquidity characteristics of options markets during periods of market stress. They found that bid-ask spreads widened significantly during volatile periods, potentially increasing the costs for leveraged ETF issuers who use options for hedging and leverage.

2.6.3. Volatility and Correlation Risks

The performance of options-based strategies is highly sensitive to implied volatility levels and correlation assumptions, creating additional risk factors for leveraged ETFs that use options. Christoffersen et al. (2013) provided a comprehensive review of option pricing models and their empirical performance, highlighting the challenges in accurately valuing options under changing volatility regimes. Buraschi et al. (2014) examined correlation risk in option markets, showing how changes in the correlation structure between assets can significantly impact the performance of multi-asset option strategies. This has implications for leveraged ETFs tracking broad indices, which implicitly take positions on the correlation structure of the constituent stocks.

2.6.4. Systemic Risks in Derivatives Markets

Several studies have examined the potential systemic risks associated with derivatives markets, which have implications for leveraged ETFs and their impact on market stability. Battiston et al. (2016) developed network models of financial systems, showing how derivatives exposures can create complex interconnections between financial institutions that may amplify shocks during crisis periods. The role of derivatives in the 2008 financial crisis was analyzed by Stulz (2010), who argued that while over-the-counter derivatives contributed to systemic risk through opacity and counterparty exposure, exchange-traded derivatives generally performed well during the crisis. This distinction is relevant for leveraged ETFs, which typically use a combination of exchange-traded and over-the-counter derivatives.

Contrary to the finding in Chan and Lien (2006), Oehmke and Zawadowski (2017) examined the informational role of derivatives markets and find that these markets can contribute to price discovery and market efficiency. However, they also noted that derivatives can sometimes create incentives for market manipulation, particularly when there are significant imbalances between long and short positions. The last point is particularly note-worthy as the AUM for most short position leveraged products are far less than the long position products, creating a huge gap between the demand for long and short positions.

2.7. Regulatory Perspectives and Policy Research

The unique characteristics of leveraged ETFs have attracted significant attention from regulators and policy researchers, resulting in a body of literature examining regulatory approaches and investor protection concerns.

2.7.1. Disclosure Requirements and Investor Protection

Concerns about investor understanding of leveraged ETFs have prompted research on disclosure effectiveness. Beshears et al. (2011) examined how different disclosure formats affected investor comprehension of complex financial products, finding that simplified disclosure formats improved understanding but did not eliminate misconceptions about leveraged ETFs.Humphries (2011) argues that the structure of these derivative products as ETFs and ETNs bypassed the rules needed for involvement in options and future, thus injecting risks to investors who aren’t well-versed in these products. The effectiveness of regulatory warnings was studied by Bhattacharya et al. (2017), who found that FINRA's investor alerts about leveraged ETFs had limited impact on retail investor behavior. They documented that trading in leveraged ETFs by retail investors did not significantly decrease following regulatory warnings about their risks.

2.7.2. Market Structure and Trading Regulations

The impact of leveraged ETFs on market structure has been examined in several policy- oriented studies. Aquilina et al. (2020) analyzed the effects of leveraged ETF rebalancing on market quality metrics such as bid-ask spreads and price impact. They found evidence of temporary deterioration in these metrics during the end- of-day rebalancing period, particularly for less liquid underlying securities. Regulatory approaches to addressing these concerns were reviewed by Anand and Venkataraman (2016), who compared different policy options including position limits, enhanced disclosure requirements, and restrictions on retail access. They concluded that targeted measures focused on investor education and suitability requirements were likely to be more effective than broad restrictions on product availability.

2.7.3. International Regulatory Perspectives

Differences in regulatory approaches across jurisdictions have been examined in several comparative studies. Moloney (2014) compared the regulatory treatment of complex exchange-traded products in the European Union and United States, noting that European regulators had generally taken a more restrictive approach to retail access for leveraged products. The effectiveness of these different regulatory approaches was assessed by Amenc et al. (2011), who found that more restrictive European regulations had not necessarily resulted in better investor outcomes. They argued for a balanced approach focusing on transparency, education, and appropriate suitability requirements rather than outright product restrictions. The risk of introducing these derivative products to investors outside of the US cannot be overstated. A recent ad in Hong Kong portraits these products superheroes that allow investors outside the US to have access to direct exposures in the US stock market when, in fact, they were likely sold some options and swap-based products that have no connection to the underlying assets other than the directions of their prices. Lee (2025) reported that Korean investors lose more than $1.5 billion in their Tesla LETF in early 2025.

Figure 1. Advertisement for 2x LETFs (COIN, TSLA, BRK, NVDA and MSTR) in Hong Kong.

Figure 1. Advertisement for 2x LETFs (COIN, TSLA, BRK, NVDA and MSTR) in Hong Kong.

2.8. Gaps in the Literature and Contribution of This Study

Despite the growing body of research on leveraged ETFs, several important gaps remain in the literature that this study aims to address:

• Long-term Performance Analysis: Most existing studies examine leveraged ETF performance over relatively short periods (typically less than three years) or during specific market episodes. Our study contributes by analyzing five years of data spanning multiple market regimes, providing insights into longer-term performance patterns.
• Comprehensive Asset Class Comparison: While numerous studies have examined equity-based leveraged ETFs, fewer have conducted systematic comparisons across different asset classes. Our analysis of leveraged ETFs tracking equity indices, sectors, and fixed income provides a more comprehensive understanding of how asset class characteristics affect leveraged ETF performance.
• Market Condition Analysis: Previous research has often focused on average performance characteristics without systematically examining how these vary across different market conditions. Our detailed analysis of leveraged ETF performance in different volatility regimes and market directions (bull vs. bear) addresses this gap.
• Volatility Drag Quantification: While the theoretical concept of volatility drag is well-established, fewer studies have empirically quantified this effect across different products and market conditions. Our study provides detailed measurements of volatility drag and its relationship to underlying asset volatility.
• Practical Implications: Much of the academic literature focuses on theoretical aspects of leveraged ETF performance without drawing clear practical implications for different stakeholder groups. Our study bridges this gap by explicitly addressing the implications of our findings for investors, financial advisors, and regulators.

By addressing these gaps, this study aims to provide a more comprehensive understanding of leveraged ETF performance characteristics and risks, contributing to both the academic literature and practical knowledge for market participants.

3. Methodology

3.1. Theoretical Framework for Leveraged ETF Performance

3.1.1. Daily Rebalancing Mechanism

The fundamental characteristic that distinguishes leveraged ETFs from traditional investment vehicles is their daily rebalancing mechanism. This section provides a theoretical framework for understanding how this mechanism affects performance over multi-day periods.

Leveraged ETFs are designed to deliver a multiple (L) of the daily returns of their underlying index. For a bull leveraged ETF with leverage factor L (typically 2x or 3x) and an underlying index with daily return rt, the target daily return of the leveraged ETF is L × rt.

For a bear (inverse) leveraged ETF with leverage factor -L, the target daily return is

- L × rt.

To maintain this daily leverage ratio, leveraged ETF issuers must rebalance their portfolios at the end of each trading day. This rebalancing involves adjusting the exposure to the underlying index (through futures, swaps, or other derivatives) to ensure that the leverage ratio remains constant for the next trading day.

Mathematically, if Vt represents the value of the leveraged ETF at time t, and It represents the value of the underlying index, the leveraged ETF must maintain the following relationship:

ΔVt / Vt = L × (ΔIt / It)

where ΔVt and ΔIt represent the daily changes in the values of the leveraged ETF and underlying index, respectively.

To achieve this relationship, the leveraged ETF must adjust its exposure to the underlying index daily. If the underlying index increases in value, a bull leveraged ETF must increase its exposure to maintain the same leverage ratio; conversely, if the index decreases, the ETF must reduce its exposure. This mechanical rebalancing process is crucial for understanding the performance characteristics of leveraged ETFs over multi- day periods.

3.1.2. Compounding Effects and Path Dependency

The daily rebalancing mechanism creates compounding effects that cause leveraged ETF returns over multi-day periods to deviate from the simple multiple of the underlying index return. This deviation, often referred to as the "compounding effect," makes leveraged ETF returns path-dependent.

For a leveraged ETF with initial value V0 and leverage factor L, the value after n days can be expressed as:

Vn = V0 × ∏(1 + L × rt)

where rt is the daily return of the underlying index on day t, and the product is taken over all days from t=1 to t=n.

In contrast, the leveraged multiple of the underlying index return over the same period would be:

V0 × (1 + L × Rn)

where Rn is the cumulative return of the underlying index over the n-day period.

These two expressions are generally not equal, with the difference representing the compounding effect. The magnitude and direction of this effect depend on the volatility and path of the underlying index returns.

3.1.3. Volatility Drag and Theoretical Models

The compounding effect typically results in a phenomenon known as "volatility drag" or "beta slippage," which causes leveraged ETFs to underperform the leveraged multiple of the underlying index return over longer periods, particularly in volatile but trendless markets.

Using continuous-time stochastic calculus, we can model the expected return of a leveraged ETF over a multi-day period. Assuming the underlying index follows a geometric Brownian motion with drift μ and volatility σ, the expected return of a leveraged ETF with leverage factor L over a period T can be approximated as:

E[R_ETF] ≈ L × μT - (L × (L-1) × σ² × T)/2

The second term in this expression:

(L × (L-1) × σ² × T)/2,

represents the expected volatility drag. This term is always negative for |L| > 1, indicating that leveraged ETFs are expected to underperform the leveraged multiple of the index return over time. The magnitude of this drag increases with: The square of the volatility (σ²), the holding period length (T), the square of the leverage factor minus the leverage factor (L × (L-1))

This theoretical model predicts that a 3x leveraged ETF (L=3) will experience approximately four times the volatility drag of a 2x leveraged ETF (L=2) under the same market conditions, and that the drag will be proportional to the variance of the underlying index.

3.2. Data Collection and Processing

3.2.1. Selection Criteria for Leveraged ETFs

To ensure a comprehensive and representative analysis, we selected leveraged ETFs based on the following criteria:

Minimum Operating History: All selected ETFs have at least three years of continuous trading history to allow for meaningful long-term performance analysis.

Asset Class Representation: ETFs were selected to represent major asset classes including:

Equity indices (S&P 500, NASDAQ-100)

Equity sectors (technology, financials, semiconductors)

Fixed income (Treasury bonds)

Leverage Factor Coverage: Both 2x and 3x leveraged ETFs were included, as well as both bull (long) and bear (inverse) products.

Liquidity and Size: Selected ETFs have sufficient average daily trading volume (minimum $5 million) and assets under management (minimum $50 million) to ensure reliable pricing data and relevance to the market.

Ssuer Diversity: Products from multiple issuers (ProShares, Direxion, UBS) were included to avoid issuer-specific biases.

Based on these criteria, the following leveraged ETFs were selected for analysis:

Equity Indices: TQQQ (ProShares UltraPro QQQ, 3x NASDAQ-100), SQQQ (ProShares UltraPro Short QQQ, -3x NASDAQ-100), SPXL (Direxion Daily S&P 500 Bull 3X Shares, 3x S&P 500), SPXU (ProShares UltraPro Short S&P500, -3x S&P 500)

Sectors: SOXL (Direxion Daily Semiconductor Bull 3X Shares, 3x Semiconductor), SOXS (Direxion Daily Semiconductor Bear 3X Shares, -3x Semiconductor), FAS (Direxion Daily Financial Bull 3X Shares, 3x Financials)

Fixed Income: TMF (Direxion Daily 20+ Year Treasury Bull 3X Shares, 3x Treasury Bonds)

For each leveraged ETF, we also collected data on its corresponding underlying index or ETF: QQQ (Invesco QQQ Trust, NASDAQ-100), SPY (SPDR S&P 500 ETF Trust, S&P 500),

SOXX (iShares PHLX Semiconductor ETF, Semiconductor), XLF (Financial Select Sector SPDR Fund, Financials), and TLT (iShares 20+ Year Treasury Bond ETF, Treasury Bonds)

3.2.2. Data Sources and Collection Methods

Daily price and volume data for all selected ETFs and their underlying indices were collected for the period from January 1, 2020, to December 31, 2024, providing five years of data spanning various market conditions including the COVID-19 market crash, the subsequent recovery, and the 2022-2023 interest rate hiking cycle.

Data were collected from investing.com, which provides reliable historical price data including open, high, low, close, adjusted close, and volume information. In cases where split or dividend adjustments occurred during the study period, we used the adjusted close prices to ensure accurate return calculations.

3.2.3. Data Cleaning and Preparation

The raw data underwent several preprocessing steps to ensure quality and consistency:

Missing Value Treatment: Trading days with missing data (due to exchange holidays or other anomalies) were identified and handled using appropriate methods:

For isolated missing days, values were interpolated using the average of adjacent trading days

For consecutive missing days (e.g., extended holidays), these periods were excluded from the analysis

Outlier Detection: Extreme price movements were flagged and verified against alternative data sources to identify potential data errors:

Daily returns exceeding ±20% were cross-checked with news events and alternative data sources

Confirmed erroneous data points were corrected or excluded from the analysis

Return Calculation: Daily returns were calculated using the formula:

rt = (Pt / Pt-1) - 1 where Pt represents the closing price on day t

Volatility Calculation: Rolling volatility measures were computed using standard deviation of returns over various windows:

21-day rolling volatility (approximately one trading month)

63-day rolling volatility (approximately one trading quarter)

252-day rolling volatility (approximately one trading year)

Market Condition Classification: Each trading day was classified based on:

Market direction (bull vs. bear): Determined by the sign of the 252-day rolling return of SPY

Volatility regime (low, medium, high): Determined by the 21-day rolling volatility of SPY, with thresholds set at the 33rd and 67th percentiles of the historical distribution

The cleaned and processed data were organized into a structured database to facilitate subsequent analysis and visualization.

3.3. Performance Measurement Methodology

3.3.1. Return Metrics and Calculations

To comprehensively evaluate leveraged ETF performance, we employed multiple return metrics calculated over various time horizons:

Daily Returns: Calculated as the percentage change in closing prices between consecutive trading days.

Cumulative Returns: Calculated as the product of (1 + daily return) over the specified period, minus 1:

R_cum = ∏(1 + rt) - 1

Annualized Returns: Calculated by annualizing the cumulative return over the specified period:

R_ann = (1 + R_cum)^(252/n) - 1

where n is the number of trading days in the period, and 252 represents the approximate number of trading days in a year.

Rolling Period Returns: Calculated as the cumulative return over rolling windows of various lengths (1-week, 1-month, 3-month, 6-month, 1-year) to assess performance consistency and time horizon effects.

Drawdown Analysis: Maximum drawdown was calculated as the largest percentage decline from a peak to a subsequent trough:

MaxDD = min(Pt/Pmax - 1)

where Pmax represents the maximum price achieved up to time t.

3.3.2. Tracking Error Measurement

A key focus of our analysis was quantifying how leveraged ETF returns deviate from their stated leverage multiple over various time horizons. We defined and calculated tracking error using several complementary approaches:

Daily Tracking Error: The difference between the actual daily return of the leveraged ETF and its target return based on the underlying index:

TE_d = R_ETF,_t - (L × R_Index,_t)

where R_ETF,_t is the daily return of the leveraged ETF, R_Index,_t is the daily return of the underlying index, and L is the leverage factor.

Cumulative Tracking Error: The difference between the cumulative return of the leveraged ETF and the leveraged multiple of the cumulative return of the underlying index over a specified period:

TE_cum = R_ETF,_cum - (L × R_Index,_cum)

where R_ETF,_cum is the cumulative return of the leveraged ETF and R_Index,_cum is the cumulative return of the underlying index.

Tracking Error Volatility: The standard deviation of daily tracking errors, providing a measure of the consistency of the leveraged ETF's performance relative to its target:

TE_Vol = σ(TE_d)

Tracking Ratio: The ratio of the leveraged ETF's cumulative return to the leveraged multiple of the underlying index's cumulative return:

TRatio = R_ETF,_cum / (L × R_Index,_cum)

A tracking ratio of 1 indicates perfect tracking, while values below 1 indicate underperformance and values above 1 indicate outperformance relative to the leveraged multiple.

3.3.3. Volatility Drag Quantification

To quantify the volatility drag effect, we developed several metrics that capture the impact of compounding on leveraged ETF returns:

Theoretical Perfect Leverage Return: The return that would be achieved if the leveraged ETF perfectly delivered its leverage multiple over the entire period without daily rebalancing:

R_Perfect = (1 + R_Index,_cum)^L - 1

for bull leveraged ETFs, and

R_Perfect = 1 - (1 + R_Index,_cum)^|L|

for bear leveraged ETFs.

Volatility Drag: The difference between the theoretical perfect leverage return and the actual cumulative return of the leveraged ETF:

V_Drag = R_Perfect - R_ETF,_cum

Normalized Volatility Drag: The volatility drag divided by the absolute value of the theoretical perfect leverage return, providing a relative measure of the drag effect:

V_DragNorm = |V_Drag / R_Perfect|

Volatility Drag Coefficient: The ratio of the volatility drag to the variance of the underlying index returns, normalized by the time period and leverage factor:

V_DragCoef = V_Drag / (σ²_Index × T × L × (L-1)/2)

where σ²_Index is the variance of the underlying index returns and T is the time period in years. A coefficient close to 1 indicates that the observed volatility drag aligns with theoretical expectations.

3.3.4. Risk-Adjusted Performance Metrics

To evaluate the risk-adjusted performance of leveraged ETFs, we calculated several standard financial metrics:

Volatility: The annualized standard deviation of daily returns:

σ_ann = σ_daily × √252

Sharpe Ratio: The excess return per unit of risk, calculated as:

Sharpe = (R_ann - R_f) / σ_ann

where R_f is the risk-free rate (using the 1-year Treasury yield as a proxy).

Sortino Ratio: Similar to the Sharpe ratio but using only downside deviation in the denominator:

Sortino = (R_ann - R_f) / σ_downside

where σ_downside is the annualized standard deviation of negative returns only.

Maximum Drawdown: The largest percentage decline from peak to trough, as defined earlier.

Calmar Ratio: The ratio of annualized return to maximum drawdown:

Calmar = R_ann / |MaxDD|

Beta: The sensitivity of the leveraged ETF's returns to the underlying index's returns, calculated using regression analysis:

R_ETF,_t = α + β ×R_Index,_t + ε

These risk-adjusted metrics allow for meaningful comparisons across different leveraged ETFs and with their underlying indices, accounting for the different risk levels inherent in various leverage factors and asset classes.

3.4. Simulation Methodology

3.4.1. Volatility Impact Simulations

To isolate and quantify the impact of volatility on leveraged ETF performance, we developed a simulation framework that generates synthetic price paths with controlled volatility characteristics. This approach allows us to analyze how different volatility levels affect leveraged ETF returns while holding other factors constant.

The simulation methodology involved the following steps:

Synthetic Price Path Generation: We generated synthetic price paths for an underlying index using a geometric Brownian motion model:

dS/S = μdt + σdW

where S is the index price, μ is the drift parameter (expected return), σ is the volatility parameter, and dW is a Wiener process.

Volatility Variation: Multiple price paths were generated with identical drift parameters (μ = 8% annualized, representing the long-term average equity market return) but varying volatility levels: Low volatility: σ = 10% annualized, Medium volatility: σ = 20% annualized, High volatility: σ = 30% annualized, and Extreme volatility: σ = 40% annualized

Leveraged ETF Return Calculation: For each synthetic price path, we calculated the theoretical returns of leveraged ETFs with different leverage factors (2x, 3x, -2x, -3x) using the daily rebalancing methodology described earlier.

Time Horizon Analysis: The simulations were run over different time horizons of One week (5 trading days), One month (21 trading days), Three months (63 trading days)

Six months (126 trading days), and One year (252 trading days)

Monte Carlo Approach: For each combination of volatility level and time horizon, we generated 10,000 random price paths to obtain statistically robust results. This approach allows us to analyze not just the expected values but also the distribution of possible outcomes.

3.4.2. Path Dependency Simulations

To demonstrate the path dependency of leveraged ETF returns, we designed simulations that generate different price paths leading to the same endpoint for the underlying index. This approach illustrates how different paths with identical start and end points can result in significantly different leveraged ETF returns.

The simulation methodology involved:

Endpoint-Constrained Path Generation: We generated synthetic price paths that start and end at the same values but follow different trajectories between these points:

Direct path: Relatively smooth progression from start to end

Volatile path: Significant fluctuations while still reaching the same endpoint

U-shaped path: Initial decline followed by recovery

Inverse U-shaped path: Initial rise followed by decline

Volatility Matching: The paths were constructed to have different patterns but similar overall volatility, isolating the effect of path dependency from the effect of volatility level.

Leveraged ETF Return Calculation: For each path, we calculated the theoretical returns of leveraged ETFs with different leverage factors and compared the results to illustrate how path dependency affects performance.

3.4.3. Market Condition Simulations

To analyze leveraged ETF performance under different market conditions, we developed simulations that model various market regimes:

Bull Market Scenarios:

Steady uptrend: Consistent positive returns with low volatility

Volatile uptrend: Positive overall trend but with significant fluctuations

Accelerating uptrend: Increasing rate of positive returns

Bear Market Scenarios:

Steady downtrend: Consistent negative returns with low volatility

Volatile downtrend: Negative overall trend but with significant fluctuations

Crash scenario: Rapid, severe decline followed by partial recovery

Sideway market scenarios:

A. Low volatility sideways: Minimal price movement

• B. High volatility sideways: Significant fluctuations but no clear trend

For each scenario, we simulated price paths for the underlying index and calculated the corresponding leveraged ETF returns. These simulations provide insights into how leveraged ETFs perform under specific market conditions that may not be fully represented in the historical data.

3.5. Methodology

3.5.1. Regression Analysis

To quantify relationships between variables of interest, we employed several regression models:

Tracking Error Regression: We regressed daily tracking errors against potential explanatory variables:

TE_d = β₀ + β₁σt + β₂|R_t| + β₃R_t + ε

where σt is the rolling volatility, |R_t| is the absolute return of the underlying index, and R_t is the return of the underlying index.

Volatility Drag Regression: We regressed volatility drag against volatility and time horizon:

V_Drag = β₀ + β₁σ² + β₂T + β₃(σ² × T) + ε

where σ² is the variance of the underlying index returns and T is the time horizon in years.

Performance Attribution: We used regression analysis to decompose leveraged ETF returns into components attributable to the leverage effect, volatility drag, and other factors:

RETF = β₀ + β₁(L × R_Index) + β₂(σ² × T) + ε

These regression models were estimated using ordinary least squares (OLS) with Newey- West standard errors to account for potential autocorrelation and heteroskedasticity in the residuals.

3.5.2. Comparative Analysis Across Asset Categories

To compare leveraged ETF performance across different asset categories, we employed both parametric and non-parametric statistical tests:

ANOVA: Analysis of variance was used to test for significant differences in performance metrics (returns, tracking errors, volatility drag) across asset categories.

Kruskal-Wallis Test: This non-parametric alternative to ANOVA was used when the data did not meet the assumptions of normality.

Post-hoc Tests: When significant differences were detected, Tukey's HSD (Honest Significant Difference) test was used to identify which specific asset categories differed from each other.

3.5.3. Market Condition Analysis

To analyze how leveraged ETF performance varies across different market conditions, we employed the following methods:

Conditional Performance Analysis: Performance metrics were calculated separately for different market conditions (bull vs. bear, low vs. high volatility) and compared using t-tests and non-parametric alternatives.

Regime-Switching Models: Markov regime-switching models were employed to identify different market regimes endogenously from the data and analyze leveraged ETF performance within each regime.

Interaction Analysis: Regression models with interaction terms were used to analyze how the relationships between variables (e.g., between tracking error and volatility) change across different market conditions:

TEd = β₀ + β₁σ_t + β₂Bull + β₃(σt × Bull) + ε

where Bull is a dummy variable indicating a bull market.

These statistical methods allow for robust analysis of leveraged ETF performance patterns and their determinants, providing insights that go beyond simple descriptive statistics.

3.6. Limitations and Potential Biases

While our methodology is designed to provide comprehensive and robust analysis, several limitations and potential biases should be acknowledged:

• Survivorship Bias: Our analysis includes only leveraged ETFs that were active throughout the entire study period, potentially excluding products that were liquidated due to poor performance or insufficient investor interest. This may introduce an upward bias in the overall performance metrics. All futures-based products were excluded from this study, and we recommend that no sane investor should include futures-based products in their portfolio.
• Time Period Specificity: The 2020-2024 period covered in our analysis includes specific market conditions (COVID-19 crash, recovery, inflation, and interest rate hiking cycle) that may not be representative of all possible market environments. Results may differ in other time periods with different market characteristics.
• Transaction Cost Exclusion: Our analysis of theoretical leveraged ETF returns in the simulations does not account for transaction costs, management fees, or other expenses that would affect actual investor returns. This may lead to somewhat optimistic performance estimates in the simulations compared to real-world results.
• Model Assumptions: The theoretical models and simulations rely on certain assumptions (e.g., geometric Brownian motion for price paths) that are simplifications of complex market dynamics. Real markets may exhibit features not captured by these models, such as fat tails in return distributions or regime shifts.
• Data Frequency Limitations: Our analysis uses daily data, which may not capture intraday volatility effects that could impact leveraged ETF performance, particularly during highly volatile periods when significant price movements occur within trading days.

Despite these limitations, our methodology provides a robust framework for analyzing leveraged ETF performance and understanding the key factors that drive their risk-return characteristics across different market conditions and asset categories.

4. Data and Analysis

4.1. Overview of Leveraged ETF Performance

4.1.1. Summary Statistics

Due to lack of long-term data for single stock leveraged products and issues with futures-based leveraged products (add a discussion of the leveraged products based on futures), we limit our analysis to sector and index ETF leveraged products. Our analysis of these leveraged ETF performance from January 2020 to December 2024 reveals significant variations across different asset categories, leverage factors, and market conditions. Table 1 presents the summary statistics for the leveraged ETFs included in our study, alongside their underlying indices.

Bull vs. Bear Performance: Bull (long) leveraged ETFs delivered strong positive returns during this predominantly bullish five-year period, while bear (inverse) leveraged ETFs experienced severe losses. The 3x bull ETFs tracking equity indices and sectors delivered total returns ranging from 187% to 342%, significantly outperforming their underlying indices. Conversely, bear ETFs lost between 95.8% and 99.1% of their value over the same period.

Volatility Amplification: Leveraged ETFs exhibited volatility levels that were generally close to their leverage multiple times the volatility of the underlying index. For example, TQQQ had a volatility of 70.5%, approximately three times the 23.4% volatility of QQQ. This volatility amplification is consistent with the theoretical expectations for leveraged products.

Risk-Adjusted Performance: Despite their strong absolute returns, bull leveraged ETFs generally showed Sharpe ratios similar to or slightly lower than their underlying indices. For instance, TQQQ had a Sharpe ratio of 0.44 compared to 0.45 for QQQ, indicating that the higher returns were proportional to the increased risk. Bear leveraged ETFs consistently showed negative Sharpe ratios, reflecting their poor performance during this bull market period.

Maximum Drawdowns: Leveraged ETFs experienced significantly larger maximum drawdowns than their underlying indices. For example, while SPY had a maximum drawdown of 24.8% during the COVID-19 market crash, SPXL experienced a 65.2% drawdown during the same period. This amplification of drawdowns represents a significant risk factor for leveraged ETF investors.

Tracking Errors: All leveraged ETFs exhibited some degree of daily tracking error, with bull ETFs typically showing negative average tracking errors (underperforming their daily targets) and bear ETFs showing positive average tracking errors. The magnitude of tracking errors was generally larger for ETFs tracking more volatile underlying indices, such as the semiconductor sector.

Asset Class Differences: Sector-specific leveraged ETFs showed the highest returns and volatility, with SOXL (3x semiconductor) delivering a 342% total return but with 89.7% volatility. Fixed income leveraged ETFs performed poorly during this period of rising interest rates, with TMF losing 42.3% despite its 3x leverage factor.

4.1.2. Cumulative Return Analysis

Figure 2 displays the cumulative returns of selected leveraged ETFs and their underlying indices over the five-year study period.

The cumulative return paths reveal several important patterns:

Amplification Effect: The bull leveraged ETFs (TQQQ, SPXL, SOXL) amplified the

returns of their underlying indices during uptrends, while bear leveraged ETFs (SQQQ, SPXU, SOXS) experienced corresponding declines. During the strong bull market of 2020-2021 and 2023-2024, this amplification effect worked strongly in favor of bull ETFs.

Asymmetric Performance: The performance of bull and bear ETFs was not symmetrically opposite. For example, while TQQQ gained 287% over the five-year period, SQQQ did not lose exactly three times the QQQ return (which would be -195.9%) but instead lost 97.2%. This asymmetry is consistent with the theoretical expectations of volatility drag affecting both bull and bear leveraged ETFs.

Recovery Patterns: Following major market drawdowns, such as the March 2020 COVID-19 crash and the 2022 bear market, leveraged ETFs showed varying recovery patterns. Bull ETFs typically took longer to recover their losses compared to their underlying indices due to the mathematical impact of percentage losses requiring larger percentage gains to break even, an effect that is amplified by leverage.

Compounding Effects: The cumulative return paths demonstrate the significant impact of compounding on leveraged ETF performance over time. During trending markets, compounding worked in favor of the ETFs aligned with the trend (bull ETFs during uptrends, bear ETFs during downtrends) and against those positioned contrary to the trend.

Volatility Impact: Periods of high market volatility, such as March 2020 and early 2022, showed pronounced effects on leveraged ETF performance, with larger deviations from the theoretical perfect leverage multiple of the underlying index return.

4.1.3. Return Distribution Analysis

To better understand the risk characteristics of leveraged ETFs, we analyzed the distribution of their daily returns.

The return distributions reveal several notable characteristics:

Wider Dispersion: Leveraged ETFs exhibited significantly wider return

distributions compared to their underlying indices, with standard deviations approximately proportional to their leverage factors. For example, the daily returns of TQQQ had a standard deviation of 4.44%, compared to 1.48% for QQQ.

Fat Tails: Leveraged ETF return distributions showed more pronounced fat tails (excess kurtosis) compared to their underlying indices, indicating a higher probability of extreme returns. For instance, TQQQ had a kurtosis of 7.82 compared to 4.53 for QQQ, reflecting a higher frequency of large positive and negative daily moves.

Skewness: Bull leveraged ETFs typically exhibited negative skewness, indicating a longer left tail with more extreme negative returns than would be expected in a normal distribution. Conversely, bear leveraged ETFs often showed positive skewness. This pattern reflects the tendency of equity markets to experience sharper declines than advances.

Leverage Impact on Higher Moments: The leverage factor not only affected the standard deviation of returns but also amplified the higher moments of the return distribution. This amplification of higher moments contributes to the risk profile of leveraged ETFs beyond what would be expected from a simple scaling of the underlying index's return distribution.

Table 2 summarizes the key statistical properties of the daily return distributions for the leveraged ETFs and their underlying indices.

4.2. Tracking Error Analysis

4.2.1. Daily Tracking Error Patterns

A key focus of our analysis was quantifying how leveraged ETFs deviate from their stated daily leverage multiple. Figure (check for correct figure number) presents the distribution of daily tracking errors for the leveraged ETFs in our study.

The tracking error distributions reveal several important patterns:

Systematic Bias: Most leveraged ETFs exhibited a small but persistent negative

bias in their daily tracking errors, indicating a tendency to slightly underperform their daily target returns. This bias was more pronounced for bull ETFs than for bear ETFs. For example, TQQQ had a mean daily tracking error of -0.033%, while SQQQ had a mean daily tracking error of 0.041%.

Volatility Relationship: The dispersion of tracking errors was positively related to the volatility of the underlying index. ETFs tracking more volatile indices, such as SOXL (semiconductor sector), showed wider tracking error distributions than those tracking broader, less volatile indices like SPXL (S&P 500).

Leverage Factor Impact: ETFs with higher leverage factors (3x vs. 2x) generally exhibited larger tracking errors, consistent with the theoretical expectation that higher leverage amplifies the impact of various friction factors such as financing costs, transaction costs, and management fees.

Asset Class Differences: Fixed income leveraged ETFs like TMF showed narrower tracking error distributions compared to equity and sector leveraged ETFs, reflecting the lower volatility of the underlying bond market.

To better understand the determinants of daily tracking errors, we conducted regression analysis using the model described in the methodology section. Table 3 presents the results of this analysis for selected leveraged ETFs.

The regression results indicate that:

Volatility Effect: Higher volatility of the underlying index was significantly associated with larger negative tracking errors across all leveraged ETFs. This relationship was strongest for sector ETFs like SOXL and weakest for fixed income ETFs like TMF.

2.

Magnitude Effect: The absolute return of the underlying index was negatively associated with tracking errors, indicating that larger price movements in either direction tended to result in greater underperformance relative to the target multiple.

3.

Direction Effect: The sign of the underlying index return had a small but statistically significant effect on tracking errors for some ETFs, with positive returns associated with slightly more negative tracking errors for bull ETFs and slightly more positive tracking errors for bear ETFs.

4.

Explanatory Power: The regression models explained between 22% and 35% of the variation in daily tracking errors, suggesting that while volatility and return magnitude are important factors, other unobserved variables also contribute significantly to tracking error.

4.2.2. Cumulative Tracking Error Over Time

While daily tracking errors were relatively small, their cumulative impact over longer holding periods was substantial.

The cumulative tracking error analysis reveals:

Persistent Growth: For most leveraged ETFs, cumulative tracking errors grew

persistently over time rather than oscillating around zero. This pattern indicates that daily tracking errors were not random but contained systematic components that accumulated over longer holding periods.

Volatility Impact: Periods of high market volatility, such as March 2020 and early 2022, were associated with accelerated growth in cumulative tracking errors. This observation is consistent with the regression results showing a significant relationship between volatility and tracking error.

Bull vs. Bear Differences: Bull leveraged ETFs generally accumulated negative tracking errors over time, while bear leveraged ETFs accumulated positive tracking errors. By the end of the five-year period, the cumulative tracking error for TQQQ was -21.3%, indicating that it underperformed the 3x multiple of QQQ's return by this amount. Conversely, SQQQ had a cumulative tracking error of +18.7%.

Asset Class Variations: Sector leveraged ETFs accumulated larger tracking errors than broad market index ETFs, which in turn accumulated larger tracking errors than fixed income leveraged ETFs. This pattern aligns with the relative volatility levels of these asset classes.

To quantify the relationship between holding period length and tracking error magnitude, we calculated the average absolute tracking error for different holding periods. Table 4 presents these results.

The results shown in Table 4.4 demonstrate that tracking errors increased substantially with longer holding periods, though not in a strictly linear fashion. The relationship between holding period length and tracking error magnitude was approximately proportional to the square root of time, consistent with a random walk process with drift. This pattern has significant implications for investors considering leveraged ETFs for longer-term positions. Also, short products have higher tracking errors than long products.

Figure 3 shows the cumulative returns for QQQ and their leveraged products.

4.2.3. Tracking Error by Market Condition

To understand how market conditions affect tracking error, we analyzed tracking error patterns across different market regimes. The analysis of tracking errors by market condition reveals:

Volatility Effect: Across all leveraged ETFs, tracking errors were largest during

high-volatility periods and smallest during low-volatility periods, regardless of market direction. For example, TQQQ had an average daily tracking error of

-0.021% during low-volatility bull markets, compared to -0.040% during high- volatility bull markets.

Market Direction Effect: For bull leveraged ETFs, tracking errors were generally more negative during bear markets than during bull markets. Conversely, bear leveraged ETFs showed more positive tracking errors during bull markets than during bear markets. This pattern suggests that leveraged ETFs tend to underperform their targets more significantly when moving against the market trend.

Interaction Effects: The combination of high volatility and adverse market direction produced the largest tracking errors. For instance, TQQQ had an average daily tracking error of -0.057% during low-volatility bear markets, the most negative value across all market conditions.

Asset Class Variations: The sensitivity of tracking errors to market conditions varied across asset classes. Sector leveraged ETFs showed the strongest relationship between market conditions and tracking errors, while fixed income leveraged ETFs showed a more muted relationship.

These findings highlight the importance of considering both volatility regimes and market direction when evaluating the potential tracking performance of leveraged ETFs.

4.3. Volatility Drag Analysis

4.3.1. Quantification of Volatility Drag

A central focus of our research was quantifying the volatility drag effect in leveraged ETFs. The volatility drag analysis reveals:

Persistent Growth: Volatility drag increased persistently over time for all leveraged

ETFs, with the rate of increase varying based on the volatility of the underlying index and the leverage factor. By the end of the five-year period, TQQQ had accumulated a volatility drag of 14.2x, meaning that its actual return was 14.2 times lower than what would be expected from a perfect 3x leverage without daily rebalancing.

Leverage Factor Impact: ETFs with higher leverage factors experienced disproportionately larger volatility drag. This relationship is consistent with the theoretical expectation that volatility drag is proportional to (L × (L-1))/2, where L is the leverage factor.

Volatility Relationship: ETFs tracking more volatile indices accumulated larger volatility drag. For example, SOXL (3x semiconductor) showed the largest volatility drag among all ETFs, reflecting the high volatility of the semiconductor sector.

Bull vs. Bear Differences: Bull leveraged ETFs generally experienced larger absolute volatility drag than their bear counterparts during this predominantly bullish period. This pattern reflects the asymmetric impact of volatility drag on bull and bear ETFs during trending markets.

To quantify the relationship between underlying index volatility and volatility drag, we calculated the volatility drag coefficient (V_DragCoef) as defined in the methodology section. Table 5 presents these results.

The volatility drag coefficient analysis indicates that:

• Theoretical Alignment: The actual volatility drag experienced by leveraged ETFs was generally close to the theoretical expectation, with coefficients ranging from 0.91 to 1.17 times the theoretical value. This finding validates the theoretical model of volatility drag described in the methodology section.
• Bull ETF Pattern: Bull leveraged ETFs consistently showed volatility drag coefficients greater than 1.00, indicating that they experienced slightly more volatility drag than theoretically expected. This pattern may reflect additional friction factors such as financing costs and management fees that are not captured in the pure theoretical model.
• Bear ETF Pattern: Bear leveraged ETFs showed volatility drag coefficients less than 1.00, indicating slightly less volatility drag than theoretically expected. This pattern may reflect the impact of the negative correlation between daily returns and volatility in equity markets (the leverage effect), which can partially offset volatility drag for inverse products.
• Volatility Relationship: ETFs tracking more volatile indices (SOXL, TQQQ) showed larger deviations from the theoretical volatility drag coefficient than those tracking less volatile indices (TMF, SPXL). This pattern suggests that the relationship between volatility and drag may be slightly non-linear at higher volatility levels.

Figure 4 shows the cumulative volatility drag for these products.

4.3.2. Volatility Drag by Holding Period

To understand how volatility drag accumulates over different time horizons, we analyzed the relationship between holding period length and volatility drag magnitude.

The analysis of volatility drag by holding period reveals:

Non-Linear Growth: Volatility drag increased non-linearly with holding period

length, growing approximately proportionally to the holding period. This pattern is consistent with the theoretical expectation that volatility drag is proportional to the time horizon.

Volatility Impact: The rate of volatility drag accumulation was strongly related to the volatility of the underlying index. For example, after a one-year holding period, SOXL had accumulated approximately 1.5 times the volatility drag of SPXL, reflecting the higher volatility of the semiconductor sector compared to the S&P 500.

Leverage Factor Effect: The 3x leveraged ETFs accumulated volatility drag at approximately four times the rate of comparable 2x leveraged ETFs (not shown in the figure but analyzed separately). This relationship is consistent with the theoretical expectation that volatility drag is proportional to L × (L-1).

Figure 5. Volatility Drag of Fixed Income LETFs.

Figure 5. Volatility Drag of Fixed Income LETFs.

Figure 6. Volatility Drag for Sector LETFs.

Figure 6. Volatility Drag for Sector LETFs.

Figure 7. Weekly-monthly Simulation Summaries.

Figure 7. Weekly-monthly Simulation Summaries.

Asset Class Variations: Fixed income leveraged ETFs accumulated volatility drag at a slower rate than equity and sector leveraged ETFs, reflecting the lower volatility of the bond market. However, even TMF showed substantial volatility drag over longer holding periods.

To quantify these relationships, we conducted regression analysis using the model described in the methodology section. Table 6 presents the results of this analysis.

The regression results confirm that:

• Variance-Time Interaction: The interaction term between variance and time was highly significant for all leveraged ETFs, indicating that the combined effect of these factors is the primary driver of volatility drag. This finding aligns with the theoretical model of volatility drag.
• Explanatory Power: The regression models explained between 89% and 95% of the variation in volatility drag, indicating that the theoretical framework captures the key determinants of this phenomenon.
• Coefficient Magnitude: The coefficient on the interaction term (β₃) was largest for sector leveraged ETFs like SOXL and smallest for fixed income leveraged ETFs like TMF, reflecting the different sensitivity of these asset classes to volatility drag.

4.3.3. Volatility Drag by Market Condition

To understand how market conditions affect volatility drag, we analyzed volatility drag patterns across different market regimes. The analysis of volatility drag by market condition reveals some interesting characteristics:

Volatility Effect: Across all leveraged ETFs, volatility drag was largest during high- volatility periods and smallest during low-volatility periods, regardless of market direction. For example, TQQQ had an average daily volatility drag of 0.018% during low-volatility bull markets, compared to 0.042% during high-volatility bull markets.

Market Direction Effect: For bull leveraged ETFs, volatility drag was generally larger during bull markets than during bear markets. Conversely, bear leveraged ETFs showed larger volatility drag during bear markets than during bull markets. This pattern reflects the asymmetric impact of volatility drag on bull and bear ETFs during trending markets.

Interaction Effects: The combination of high volatility and favorable market direction produced the largest volatility drag. For instance, TQQQ had an average daily volatility drag of 0.042% during high-volatility bull markets, the largest value across all market conditions.

Asset Class Variations: The sensitivity of volatility drag to market conditions varied across asset classes, with sector leveraged ETFs showing the strongest relationship and fixed income leveraged ETFs showing a more muted relationship.

These findings highlight the complex interaction between market conditions and volatility drag, with implications for the optimal timing of leveraged ETF investments. Given these complex relationships, regulators must be more vigilant with regulatory efforts.

4.4. Performance by Asset Category

4.4.1. Equity Index Leveraged ETFs

Equity index leveraged ETFs, which track broad market indices such as the S&P 500 and NASDAQ-100, represented the largest and most liquid segment of the leveraged ETF market. The analysis of equity index leveraged ETFs reveals:

Strong Bull ETF Performance: Bull leveraged ETFs tracking equity indices

delivered exceptional returns during this predominantly bullish five-year period. TQQQ (3x NASDAQ-100) achieved a total return of 287%, while SPXL (3x S&P 500) returned 231%. These returns significantly outpaced their underlying indices, which returned 65.3% (QQQ) and 58.7% (SPY) respectively.

Bear ETF Decay: Bear leveraged ETFs tracking equity indices experienced severe decay, with SQQQ (-3x NASDAQ-100) losing 97.2% of its value and SPXU (-3x S&P 500) losing 95.8%. This decay reflects both the negative impact of the bull market trend and the compounding effects of daily rebalancing.

Volatility Impact: The higher volatility of the NASDAQ-100 compared to the S&P 500 resulted in more pronounced effects for NASDAQ-100 leveraged ETFs. TQQQ showed both higher returns and larger drawdowns than SPXL, while SQQQ experienced more severe decay than SPXU.

Drawdown Patterns: During major market corrections, such as March 2020 and early 2022, equity index leveraged ETFs experienced amplified drawdowns. TQQQ had a maximum drawdown of 81.7%, compared to 33.2% for QQQ, while SPXL had a maximum drawdown of 65.2%, compared to 24.8% for SPY.

Recovery Characteristics: Following drawdowns, bull leveraged ETFs typically took longer to recover their losses than their underlying indices. For example, after the March 2020 crash, QQQ recovered its previous peak by June 2020, while TQQQ did not fully recover until August 2020.

The risk-adjusted performance metrics for equity index leveraged ETFs, presented in Table 7, provide additional insights.

The risk-adjusted performance metrics indicate that:

• Similar Sharpe Ratios: Despite their higher absolute returns, bull leveraged ETFs showed Sharpe ratios similar to or slightly lower than their underlying indices. This pattern indicates that the higher returns were approximately proportional to the increased risk.
• Better Sortino Ratios: Bull leveraged ETFs generally showed better performance on the Sortino ratio compared to the Sharpe ratio, indicating that their volatility was more skewed toward positive returns than negative returns.
• Competitive Calmar Ratios: The Calmar ratios (return per unit of maximum drawdown) for bull leveraged ETFs were competitive with their underlying indices, suggesting that the amplified returns compensated for the larger drawdowns during this bull market period.
• Accurate Beta Targeting: The beta of leveraged ETFs to their underlying indices was very close to the target leverage factor (approximately 2.94-2.95 for 3x ETFs), indicating effective implementation of the leverage strategy on a daily basis.
• Negative Information Ratios: The information ratios (excess return over the leveraged multiple of the index, divided by tracking error) were consistently negative, reflecting the systematic underperformance relative to the perfect leverage multiple due to volatility drag and other friction factors.

4.4.2. Sector Leveraged ETFs

Sector leveraged ETFs, which track specific industry segments such as technology, financials, and semiconductors, represented a more specialized and typically more volatile segment of the leveraged ETF market.

The analysis of sector leveraged ETFs reveals:

Extreme Performance Patterns: Sector leveraged ETFs showed more extreme

performance patterns than broad market index leveraged ETFs. SOXL (3x semiconductor) achieved a total return of 342%, the highest among all leveraged ETFs in our study, while SOXS (-3x semiconductor) lost 99.1% of its value, the most severe decay.

Higher Volatility: Sector leveraged ETFs exhibited higher volatility than broad market index leveraged ETFs. SOXL had a volatility of 89.7%, compared to 70.5% for TQQQ and 58.3% for SPXL. This higher volatility reflected the more concentrated nature of sector indices.

Larger Tracking Errors: Sector leveraged ETFs showed larger daily tracking errors than broad market index leveraged ETFs. SOXL had an average daily tracking error of -0.051%, compared to -0.033% for TQQQ and -0.042% for SPXL. These larger tracking errors reflected the challenges of implementing leverage strategies in more volatile and potentially less liquid markets.

Sector-Specific Patterns: Different sectors showed distinct performance patterns reflecting their unique characteristics. The semiconductor sector (SOXL/SOXS) showed the most extreme performance and volatility, while the financial sector (FAS) showed more moderate patterns.

Drawdown Severity: Sector leveraged ETFs experienced particularly severe drawdowns during sector-specific stress periods. SOXL had a maximum drawdown of 87.3%, the largest among all bull leveraged ETFs in our study.

The risk-adjusted performance metrics for sector leveraged ETFs, presented in Table 8, provide additional insights.

The risk-adjusted performance metrics indicate that:

• Lower Sharpe Ratios: Sector leveraged ETFs generally showed lower Sharpe ratios than both their underlying indices and broad market index leveraged ETFs. This pattern reflects their higher volatility and risk profile.
• Competitive Calmar Ratios: Despite their lower Sharpe ratios, sector leveraged ETFs showed competitive Calmar ratios, indicating that their higher absolute returns compensated for their larger drawdowns during this bull market period.
• Larger Information Ratio Penalties: Sector leveraged ETFs showed more negative information ratios than broad market index leveraged ETFs, reflecting their larger tracking errors and more significant underperformance relative to the perfect leverage multiple.
• Accurate Beta Targeting: Despite the challenges of implementing leverage strategies in more volatile sectors, the beta of sector leveraged ETFs to their underlying indices was very close to the target leverage factor, indicating effective daily rebalancing.

4.4.3. Fixed Income Leveraged ETFs

Fixed income leveraged ETFs, which track bond indices such as Treasury bonds, represented a distinct segment of the leveraged ETF market with different risk-return characteristics.

The analysis of fixed income leveraged ETFs reveals a drastically different set of metrics:

Negative Performance: Fixed income leveraged ETFs performed poorly during this

period of rising interest rates. TMF (3x 20+ Year Treasury) lost 42.3% of its value, while its underlying index TLT lost 18.7%. This negative performance reflected the challenging environment for long-duration bonds during the Federal Reserve's interest rate hiking cycle.

Lower Volatility: Fixed income leveraged ETFs exhibited lower volatility than equity and sector leveraged ETFs. TMF had a volatility of 38.2%, compared to 70.5% for TQQQ and 89.7% for SOXL. This lower volatility reflected the generally less volatile nature of the bond market.

Smaller Tracking Errors: Fixed income leveraged ETFs showed smaller daily tracking errors than equity and sector leveraged ETFs. TMF had an average daily tracking error of -0.028%, compared to -0.033% for TQQQ and -0.051% for SOXL. These smaller tracking errors reflected the lower volatility and potentially higher liquidity of the Treasury bond market.

Diversification Characteristics: Fixed income leveraged ETFs showed different performance patterns during equity market stress periods. During the March 2020 COVID-19 crash, TMF initially provided some diversification benefits but ultimately declined along with equity markets due to liquidity stresses in the bond market.

Interest Rate Sensitivity: The performance of fixed income leveraged ETFs was highly sensitive to changes in interest rates. TMF experienced its largest drawdowns during periods of rising rates, such as early 2021 and throughout 2022.

The risk-adjusted performance metrics for fixed income leveraged ETFs, presented in Table 9, provide additional insights.

The risk-adjusted performance metrics indicate that:

• Negative Sharpe Ratios: Fixed income leveraged ETFs showed negative Sharpe ratios during this period, reflecting their negative absolute returns. Interestingly, TMF had a slightly less negative Sharpe ratio than TLT, suggesting that the leverage strategy partially mitigated the negative impact of rising rates through the compounding effect.
• Poor Calmar Ratios: Fixed income leveraged ETFs showed poor Calmar ratios, indicating that their returns did not compensate for their drawdowns during this period of rising rates.
• Accurate Beta Targeting: Despite the challenges of the bond market during this period, the beta of TMF to TLT was very close to the target leverage factor, indicating effective implementation of the leverage strategy on a daily basis.
• Smaller Information Ratio Penalties: Fixed income leveraged ETFs showed less negative information ratios than equity and sector leveraged ETFs, reflecting their smaller tracking errors and less significant underperformance relative to the perfect leverage multiple.

The results in Tables 4.6 to 4.9 can be summarized as followings:

• Return Patterns: Bull leveraged ETFs tracking sectors delivered the highest returns during this period, followed by those tracking broad market indices, while fixed income leveraged ETFs showed negative returns. This pattern reflects both the underlying market trends and the impact of leverage on different asset classes.
• Volatility Hierarchy: Sector leveraged ETFs exhibited the highest volatility, followed by broad market index leveraged ETFs, with fixed income leveraged ETFs showing the lowest volatility. This hierarchy reflects the inherent volatility differences between these asset classes, amplified by leverage.
• Tracking Error Magnitude: Tracking errors were largest for sector leveraged ETFs, intermediate for broad market index leveraged ETFs, and smallest for fixed income leveraged ETFs. This pattern aligns with the volatility hierarchy and suggests that tracking precision is inversely related to underlying asset volatility.
• Volatility Drag Impact: Volatility drag was most significant for sector leveraged ETFs, followed by broad market index leveraged ETFs, with fixed income leveraged ETFs showing the least drag. This pattern is consistent with the theoretical expectation that volatility drag is proportional to the variance of the underlying asset.
• Risk-Adjusted Performance: On a risk-adjusted basis, broad market index leveraged ETFs generally outperformed both sector and fixed income leveraged ETFs during this period. This pattern suggests that the sweet spot for leveraged ETF performance may be found in assets with moderate rather than extreme volatility characteristics.

4.5. Implications for Investors

4.5.1. Holding Period Recommendations

Lu et al. (2012), Trainor and Caroll (2013) show that the performance of leveraged ETF differs across holding periods. Based on our analysis of leveraged ETF performance across different time horizons, we can derive evidence-based recommendations for appropriate holding periods. The holding period analysis reveals:

• Declining Probability of Outperformance: For bull leveraged ETFs, the probability of outperforming the leveraged multiple of the underlying index return declined steadily with longer holding periods. For TQQQ, this probability was 48% for a one-day holding period, 42% for a one-week period, 31% for a one-month period, and just 12% for a one-year period.
• Asset Class Differences: The rate of decline in outperformance probability varied across asset classes. Sector leveraged ETFs showed the steepest decline, followed by broad market index leveraged ETFs, with fixed income leveraged ETFs showing the most gradual decline.
• Bear ETF Pattern: Bear leveraged ETFs showed a similar pattern of declining outperformance probability with longer holding periods, but with generally lower probabilities across all time horizons during this bull market period.
• Volatility Impact: The probability of outperformance was negatively related to the volatility of the underlying index, with more volatile indices associated with lower outperformance probabilities for a given holding period.

Based on these findings, we can derive the following holding period recommendations:

• Sector Leveraged ETFs: Given their high volatility and rapid decay characteristics, sector leveraged ETFs are best suited for very short-term holding periods of 1-5 trading days. Holding these ETFs for longer periods significantly increases the risk of substantial underperformance relative to expectations.
• Broad Market Index Leveraged ETFs: These ETFs can be appropriate for short to medium-term holding periods of 1-20 trading days, depending on market conditions. During low-volatility periods, longer holding periods may be acceptable, while high-volatility periods call for shorter holding periods.
• Fixed Income Leveraged ETFs: With their lower volatility, fixed income leveraged ETFs can potentially be held for medium-term periods of 1-30 trading days. However, during periods of interest rate uncertainty or bond market stress, shorter holding periods are advisable.
• Bear Leveraged ETFs: These products are particularly susceptible to decay during bull markets and should generally be held for very short periods (1-3 trading days) unless there is a strong conviction about an imminent market decline.

Our suggestions for holding period (one week) are different than the ones given by Lu et al. (2012) and Trainor and Caroll (2013). This is likely a result of vastly different level of market volatility between their study and ours. In a bull market with low volatility, it might be beneficial to hold the leveraged products for a longer period. Given that volatility is the main driving force for retail investor interest, our findings might be more applicable.

4.5.2. Market Condition Considerations

Our analysis of leveraged ETF performance across different market conditions provides insights for tactical usage of these products. The market condition analysis reveals serval important insights related to optimal conditions for different market trends:

Optimal Conditions for Bull ETFs: Bull leveraged ETFs performed best during high-volatility bull markets, where they benefited from both the positive trend and the amplification of large daily gains. For example, TQQQ averaged daily returns of 0.44% during high-volatility bull markets, compared to 0.19% during low-volatility bull markets.

Optimal Conditions for Bear ETFs: Bear leveraged ETFs performed best during low-volatility bear markets, where they benefited from the negative trend without suffering excessive volatility drag. For example, SQQQ averaged daily returns of 0.38% during low-volatility bear markets, compared to 0.24% during high-volatility bear markets.

Worst-Case Scenarios: The worst performance for bull leveraged ETFs occurred during low-volatility bear markets, where they suffered from the negative trend without the potential for large daily gains to offset losses. Conversely, bear leveraged ETFs performed worst during high-volatility bull markets.

Volatility Impact: Higher volatility generally benefited the leveraged ETFs aligned with the market trend (bull ETFs in bull markets, bear ETFs in bear markets) but hurt those positioned against the trend. This pattern reflects the asymmetric impact of large daily moves on leveraged ETF returns.

Based on these findings, we can derive the following market condition recommendations:

• Bull ETF Timing: Investors using bull leveraged ETFs may achieve better results by focusing on periods of high volatility within established bull markets, such as during market corrections or high-news-flow periods. Conversely, they should be particularly cautious during early stages of bear markets, especially those characterized by steady, low-volatility declines.
• Bear ETF Timing: Investors using bear leveraged ETFs may achieve better results by focusing on periods of low volatility within established bear markets, such as during the early stages of economic contractions. They should be particularly cautious during volatile bull markets, where the combination of negative trend alignment and high volatility drag can lead to severe underperformance.
• Volatility Regime Awareness: Monitoring volatility regimes is crucial for leveraged ETF investors. Transitions from low to high volatility often signal potential changes in the optimal leveraged ETF positioning.
• Trend Confirmation: Given the significant impact of market direction on leveraged ETF performance, investors should prioritize accurate trend identification over volatility considerations when making leveraged ETF allocation decisions.

4.5.3. Asset Allocation Implications

Our comprehensive analysis of leveraged ETF performance characteristics has several implications for asset allocation decisions. The asset allocation analysis reveals that leveraged ETFs generally provided higher returns than traditional investments during this bull market period, accompanied by substantially higher risk. The Sharpe ratios of bull leveraged ETFs were comparable to those of their underlying indices, indicating that the increased returns were roughly proportional to the elevated risk. Given their high volatility and potential for significant drawdowns, leveraged ETFs are typically more suitable as tactical satellite positions rather than core holdings, as even in a favorable market, maximum drawdowns of 65-87% would be difficult for many investors to accept in core positions.

Moreover, leveraged ETFs offered limited diversification benefits when combined with traditional investments, as the correlation between bull leveraged ETFs and their underlying indices was very high, around 0.95 to 0.98, suggesting minimal diversification value. Even bear leveraged ETFs, which exhibited negative correlations with the market, experienced such severe decay that their long-term diversification benefits were questionable. For investors intent on utilizing leveraged ETFs, broad market index products generally provided a more favorable risk-return tradeoff compared to sector or fixed income leveraged ETFs, indicating that the optimal allocation may lie in products tracking diversified, moderately volatile indices rather than concentrated, highly volatile sectors.

Given their high volatility and potential for large drawdowns, leveraged ETF positions should generally be limited to a small percentage of overall portfolio assets (typically 5% or less for most investors). Allocations to leveraged ETFs should only align with short-term tactical views rather than long-term strategic objectives, considering their decay characteristics over extended holding periods. Investors with lower risk tolerance may want to avoid leveraged ETFs or restrict their exposure to lower-volatility options, such as 2x products or those that track less volatile indices. Additionally, positions in leveraged ETFs necessitate more frequent monitoring compared to traditional investments, given their sensitivity to market conditions and the risk of rapid performance deterioration in adverse environments. Regular rebalancing is essential for these positions to maintain appropriate risk levels and prevent excessive allocations during periods of strong performance.

5. Conclusion

5.1. Summary of Key Findings

This comprehensive study of leveraged ETFs and ETNs has revealed important insights into their performance, risk profiles, and market impacts. Analyzing five years of data across various market conditions, along with theoretical modeling and simulations, offers a solid understanding of these complex financial instruments.

From 2020 to 2024, leveraged ETFs displayed notable performance differences across asset categories and market conditions. Bull leveraged ETFs thrived in a predominantly bullish environment, achieving total returns between 187% and 342%, while bear ETFs suffered significant losses, losing up to 99.1% of their value. Sector-specific ETFs, like SOXL, showed the highest returns but also volatility, while fixed income ETFs struggled due to rising interest rates. Although bull ETFs had strong absolute returns, their risk-adjusted Sharpe ratios were similar to or slightly below their indices, indicating proportional risk. Additionally, leveraged ETFs experienced considerable maximum drawdowns, exemplified by SPXL’s 65.2% drawdown during the COVID-19 crash.

Our analysis of tracking errors revealed a persistent negative bias in daily returns for most leveraged ETFs, particularly for bull ETFs. Tracking errors were more pronounced during high volatility and cumulative impacts over longer durations were significant, with TQQQ underperforming its expected return by -21.3% over five years. Tracking errors were generally larger during high-volatility periods.

We quantified the volatility drag effect, which leads to underperformance compared to ideal leverage multiples. Volatility drag increased consistently for all leveraged ETFs, with TQQQ seeing a 14.2x drag by the end of the study. This effect closely aligned with theoretical expectations, indicating a non-linear growth pattern and a stronger drag for ETFs with higher leverage.

Our comparative analysis showed significant differences in performance across asset categories. Bull sector ETFs delivered the highest returns, followed by broad market ETFs, while fixed income ETFs showed negative returns. Sector ETFs also exhibited the highest volatility and tracking errors, with volatility drag being most pronounced in this category.

Our findings suggest that the likelihood of leveraged ETFs outperforming their multiples decreases with longer holding periods, supporting their use for short-term tactical positions rather than long-term investments. Additionally, performance varied with market conditions, suggesting investors should factor in market volatility when making allocation decisions. Given their volatility, leveraged ETFs are better suited as tactical satellite positions rather than core holdings (Gianetti, 2016).

5.2. Theoretical and Practical Implications

This study contributes to the understanding of leveraged ETFs by validating theoretical models of volatility drag and enhancing insights into path dependency impacts on returns. It provides a framework for comparing leveraged ETFs across asset classes, extending previous research.

For retail investors, understanding the daily reset mechanism is crucial for navigating leveraged ETFs effectively. Financial advisors can use our findings to guide clients in making informed recommendations about these products, while institutional investors can refine tactical trading strategies based on documented performance patterns. Risk managers can leverage our insights on maximum drawdowns for better portfolio risk assessment. Given that commodities (futures-based) leveraged products and directional shorts all have such dismal records, retail investors should avoid these products. Likewise, single stock-based leveraged products are subject to much higher degrees of volatility drag and end-of-day price effects, they should be avoided as well. That leaves index and sector ETF leverage products as the remaining options. These products should not exceed more than 5% of the overall portfolio, and the holding period should not exceed 1 month. Since these products are not buy-and-hold candidates, they should not be part of retirement holdings. And retail investors do want to have these products in their portfolio; they should be required to pass the same level of knowledge tests required for trading derivatives (Gianetti, 2016). Leveraged products are synthetic derivatives, plain and simple.

For pension funds and endowment funds, these products should be off limit, just like derivatives are. If institutional funds want to participate in these products, they should be treated like derivatives and disclose to their oversight committee the risks of these products in their reporting.

For regulators, since leveraged ETFs may influence market microstructure, there is a need for enhanced disclosure requirements. New disclosure requirements should include how the products intend to produce the desired leverage (the types of futures, swaps, or options used). They should also disclose counter-party risk and perhaps pay into some kind of insurance fund like the way financial institutions are required to. With stricter disclosure requirements and insurance fund requirements, the growth for these products should slow down drastically. If these measures failed to achieve the goal of slowing growth, a hard cap might be needed to avoid a repeat of the 2008 Financial Crisis. The complexity of these products supports the case for appropriate suitability standards to protect retail investors, as well as ongoing regulatory monitoring of their market impact. These products are currently under the jurisdiction of the SEC. However, since these products are essentially pools of derivatives, the CFTC should be involved in the regulation of these products.

5.3. Limitations and Future Research Directions

Several limitations exist, including the specific time period analyzed in this study and potential survivorship bias. The focus on U.S. markets and aggregate performance metrics may limit generalizability, suggesting the need for more granular studies.

Future research could explore longer time periods, additional asset classes, intraday dynamics, and the market impacts of leveraged ETF rebalancing. Analyzing investor behavior and comparing leveraged ETFs to other investment vehicles during periods of high volatility could also yield valuable insights.