Margin call risk and leverage constraints: exploring investment horizons and low-risk anomalies in futures markets

Leverage constraints are substantially present in practice, limiting the ability of investors to leverage their positions adequately ^[1]. One significant consequence of leverage constraints is the emergence of low-risk anomalies in financial markets (Black, 1972; Black et al., 1972; Baker et al., 2011; Asness et al., 2013a; Frazzini and Pedersen, 2014; Boguth and Simutin, 2018; Alquist et al., 2020). Low-risk anomalies refer to the phenomenon such that low beta assets outperform high beta assets, challenging the modern asset pricing theory that associates high risks with high returns. In other words, under leverage constraints, investors tend to favor high-beta assets over low-beta assets in their quest for higher returns in financial markets.

In terms of leverage constraints, compared to other financial markets, futures markets have a unique feature called margin trading, allowing investors to utilize high levels of leverage. Margin trading enables investors to trade contracts with a margin requirement typically ranging between 3% and 12% of the notional value of futures contracts, expanding their positions up to 10–30 times. Thus, leverage constraints may not be as stringent in futures markets as they are in other markets ^[2]. For example, numerous studies provide compelling evidence that highly leveraged investors are actively engaging in futures markets (Daskalaki and Skiadopoulos, 2016; Heimer and Simsek, 2019; Ladley et al., 2020; Subrahmanyam et al., 2024). However, Frazzini and Pedersen (2014) argue for the presence of low-risk anomalies in futures markets, suggesting that investors might be leverage-constrained in these markets. This study presents evidence of such anomalies through the “Betting Against Beta (BAB)” portfolio, which involves taking long positions in low-beta futures and short positions in high-beta futures. Their empirical analysis provides evidence of low-risk anomalies for the combined portfolio across all futures sectors, although the results are inconsistent within individual sectors. Except for the equity indices sector, there is no evidence of low-risk anomalies in the commodity, country bonds, or foreign exchange futures sectors. Thus, despite substantial evidence supporting the feasibility of high-leverage trading via margin trading and the significant number of investors engaging in such practices, the findings of Frazzini and Pedersen (2014) raise the question that futures markets can be subject to leverage constraints. Moreover, the lack of consistent and significant evidence of low-risk anomalies in individual futures sectors heightens the uncertainty of the leverage constraints in futures markets.

We investigate leverage constraints in futures markets with a focus on margin trading. While margin trading typically relaxes leverage constraints for investors, it may act as a leverage constraint under certain conditions. In this study, we propose that margin call risk plays a significant role in determining leverage constraints in futures markets. As investors receive a margin call when the losses incurred from margin trading exceed their maintenance margin, the broker or exchange managing the trade will automatically liquidate the investors’ position unless they immediately add funds to bring their balance back to the required maintenance margin level. Consequently, faced with a high possibility of a margin call, investors may adopt a more cautious approach to margin trading and choose to reduce their leverage. Therefore, the risk of margin calls can impose significant leverage constraints in futures markets. From this perspective, Frazzini and Pedersen (2022) distinguish futures contracts from other assets that provide easy leverage, such as leveraged ETFs and options, by highlighting the inclusion of potential margin call risk ^[3].

The literature on margin call risk examines the extent of margin trading in futures markets and how margin call risk has different effects on investor performance and forced liquidation, particularly focusing on investor skill heterogeneity (Ladley et al., 2020; Subrahmanyam et al., 2024). Other studies have examined the relationship between margin call risk and financial crises (Brunnermeier and Pedersen, 2009; Ben-David et al., 2012; Foley et al., 2022), as well as how changes in margin call risk affect the trading performance after implementing leverage constraint policy (Heimer and Simsek, 2019; Heimer and Imas, 2022). One of our relevant studies attempted to recognize the risk of leveraged portfolios, including margin call risk, by incorporating a leverage aversion term in the mean-variance utility function to account for the inherent risk associated with margin trading (Jacobs and Levy, 2012, 2013). However, there are few studies that directly incorporate margin call risk into models. In this study, we directly incorporate margin call risk into the simulated distributions of leveraged returns based on historical datasets. This method can simplify the understanding of margin trading by providing a simple and direct means of accounting for the risk inherent in margin calls in futures trading.

Our study shows a robust and positive relationship between investment horizon and margin call risk, regardless of the level of leverage (initial margin) or the presence of a maintenance margin, as evidenced by bootstrap simulation analyses. Additionally, findings from applying a mean-variance model indicate that investors adjust their leverage in response to margin call risks, exhibiting a preference for reduced leverage over extended holding periods. Furthermore, our analysis of BAB portfolios emphasizes the importance of the holding period in determining the presence of low-risk anomalies.

Specifically, we first define the margin call risk based on the first-hitting problem of a stochastic process for the analyses. We then measure how margin call risk changes with key attributes of margin trading, including the initial margin (leverage ratio), maintenance margin, and investment horizon ^[4]. Following the approach of Fama and French (2018a, b), we use a bootstrap simulation, a widely used statistical technique, with historical data from futures markets. This method enables us to comprehensively explore the implications of margin call risk across various scenarios. We generate 100,000 bootstrap sample paths for a specific set of conditions for all individual futures, and within each simulated distribution, a margin call event occurs when the loss of a simulated cumulative excess return path exceeds the maintenance margin. We measure the margin call risk and the probability of receiving a margin call (also referred to as “margin call probability” hereinafter) and examine how the margin call risk changes with different investment horizons and initial and maintenance margin levels ^[5].

We then analyze investors’ decision-making processes to determine their optimal leverage, considering the margin call risk. To investigate how investors adjust their leverage levels in response to changing margin call risks, we use a mean-variance model by Markowitz (1952) while incorporating investor risk aversion rates. From the analyses, we identify the optimal leverage ratio that maximizes investors’ expected utility based on the simulated distribution of leveraged excess returns. This analysis enables us to understand the underlying behavioral changes of investors that result in the presence of low-risk anomalies in response to margin call risk. Finally, we conduct BAB analyses to identify the presence of low-risk anomalies across various investment horizons, which are key variables in determining margin call risk.

Our analysis reveals that margin call risk is highly responsive to changes in investment horizons across all individual futures. We find that margin call risk increases with longer investment horizons and lower initial margin levels. Interestingly, even with a low initial margin (i.e. a high leverage ratio), the likelihood of encountering a margin call remains minimal for very short holding periods (e.g. one week). However, for longer investment horizons, a small increase in leverage results in a higher mean margin call probability. Moreover, the cross-sectional difference in margin call probability diminishes as the investment horizon expands. Overall, the results provide empirical evidence that the investment horizon offers a clear basis for the level of margin call risk. In addition, we find that investors become more cautious and prefer lower leverage levels when facing substantial margin call risk over longer investment horizons. Specifically, there is a significant decrease in optimal leverage ratios when the investment horizon extends beyond a certain period, such as three months. However, for investment horizons shorter than three months, the optimal leverage ratios remain relatively high, indicating a preference for higher leverage in the absence of significant margin call risk.

Based on the above findings, a scenario favoring high-beta futures could be expected when the investment horizon is sufficiently long, as leverage constraints might arise due to the significant margin call risk. Our analysis of the BAB portfolios shows that the BAB return becomes significant and strong as the holding period increases, confirming that the presence of low-risk anomalies depends on the level of margin call risk. In particular, once the holding period exceeds six months, the BAB return reaches statistical significance at the 10% level across all sectors. The significance level further increases to approximately 1% as the holding period exceeds one year. Furthermore, when the holding period is shorter than six months, the BAB return becomes insignificant at any significance level in all sectors, with the sole exception of the equity sector. Again, the results provide supportive evidence regarding the impact of margin call risk on leverage constraints in futures markets and investors’ decisions on the use of leverage.

Many studies have attempted to explain low-risk anomalies, and the most relevant to our research are those focusing on leverage constraints (Black, 1972; Frazzini and Pedersen, 2014; Baker et al., 2011; Bali et al., 2011; Hong and Sraer, 2016; Bali et al., 2017; Boguth and Simutin, 2018; Jylhä, 2018; Schneider et al., 2020; among others). Black (1972) initially aimed to reconcile a flat security market line (SML) by relaxing one of the central CAPM assumptions about borrowing at the risk-free rate. Building on Black (1972)’s concept, Frazzini and Pedersen (2014) suggest that borrowing restrictions influence the shape of the SML. They introduce a model in which leverage-constrained investors drive up the prices of high-beta assets, resulting in lower risk-adjusted returns. More recently, Boguth and Simutin (2018) showed that the mean market beta of actively managed mutual funds reflects their desire for leverage and, consequently, the level of constraint tightness. Jylhä (2018) further supports the role of leverage constraints by demonstrating the connection between the SML’s slope and margin requirements.

We enrich the existing literature by examining how the level of leverage constraints changes and, thus, how low-risk anomalies emerge within the context of high-leverage capable markets. Our study provides empirical evidence that all distinct sectors in futures markets exhibit statistically significant low-risk anomalies under leverage constraint conditions, which preferentially depend on the investment period. Our findings imply that investors typically do not need to consider low-risk anomalies (i.e. BAB) as a systematic risk factor in futures markets. However, when the margin call risk is high, such as for sufficiently long investment horizons, investors should indeed consider low-risk anomalies as a potential systematic factor ^[6].

We further contribute to the extensive body of research on margined trades and margin call risk. Many studies have investigated the influence of margin changes on price, volatility, and liquidity. Concerning margin call risk, previous studies have focused on how margin call risk or forced liquidation triggered and exacerbated recent financial crises, such as the U.S. financial crisis of 2008–2009 (Brunnermeier and Pedersen, 2009; Ben-David et al., 2012) and the COVID-19 period (Foley et al., 2022). On the other hand, Heimer and Simsek (2019) and Heimer and Imas (2022) show that reducing the margin call risk through a leverage constraint policy in the US forex market reduces high-leverage traders’ losses by 40% and results in better trading performance. Notably, Ladley et al. (2020) and Subrahmanyam et al. (2024) analyze account-level futures trading data from Chinese futures markets and demonstrate that managing margin call risk is a key factor in investor performance. Subrahmanyam et al. (2024) show that for an average investor using a leverage ratio of 10x, the daily probability of suffering forced liquidation is 1.29%. For skilled (unskilled) investors using leverage of approximately 11x (10x), the probability of forced liquidations is 0.51% (1.74%). At high levels of leverage, the difference in the probability of forced liquidation between skilled and unskilled investors is statistically significant. Building on the importance of managing margin call risk in highly leveraged markets, our study extends the analysis by investigating the probability of margin calls and optimal leverage across various investment horizons using simulated methods in global futures markets.

Finally, our research is closely related to existing studies on how the risk of margin trading influences investor decision-making processes. For example, studies such as Jacobs and Levy (2012, 2013) try to address the limitations of mean-variance optimization models that do not explicitly incorporate margin call risk by introducing a leverage aversion term in the mean-variance utility function to account for the inherent risks associated with margin trading. Consequently, investors in these models exhibit a preference for highly leveraged portfolios. In contrast, our approach does not require the inclusion of additional assumptions or terms in the mean-variance optimization framework to capture margin trading and margin call risk. Instead, we directly incorporate margin call risk into the simulated distributions of leveraged returns. Our methodology simplifies the understanding of margin trading by providing a simple and direct means of accounting for the risks inherent in margin calls in futures trading.

The remainder of this paper is organized as follows: Section 2 describes the important features of margin trading and margin call risk, Section 3 describes our data and methodology, and Section 4 presents our results. Section 5 provides conclusions.

In this section, we describe the data used to construct future excess returns. We then provide details of a bootstrap simulation process for the leveraged futures excess return distribution and the margin call probability. In addition, we describe how we identify the optimal leverage ratio of investors according to the margin call risk. Finally, we explain the analysis of the BAB portfolios.

In this study, we use two individual datasets. First, we use a dataset from Datastream, which covers the period from January 1973 to June 2019 and includes 9 equity index futures, 13 government bond futures in developed markets, 31 commodity futures, and 7 currency futures. Summary statistics of all 60 individual futures excess returns are provided in  Table A1.

To accurately capture the dynamics of futures markets, we focus on daily closing prices across four asset classes: equities, bonds, commodities, and currencies. Because of the prevalent daily mark-to-market settlement mechanism that triggers margin calls on a daily basis in futures markets, we use daily data to compute daily excess returns rather than weekly or monthly data, following the methodology of Moskowitz et al. (2012). The use of daily data enables us to effectively analyze the frequency of margin calls and their impact.

In addition, to calculate the most liquid daily excess returns while avoiding issues that may occur after the first notice date, we use the nearest non-expiring futures contract within the specific month for which the excess return is being calculated. This approach follows the methodology of Gorton et al. (2013) and Bakshi et al. (2019).

We define margin call risk as the probability of receiving a margin call based on the first-hitting problem of a stochastic process. Following Ladley et al. (2020), a margin call occurs when the future price $Ft$ falls below the price $Fm$ at which the investor leverage loss through margin trading exceeds the maintenance margin. We can obtain the probability of a margin call $P(Ft≤Fm)$ as follows:

where $Ft$ is the future price at $t$⁠, $Fm$ is the price exceeding the maintenance margin, and $σ$ is the volatility term of a standard Wiener process.

Based on Equation (1), the probability of a margin call $P(Ft≤Fm)$ increases with both volatility $σ$ and the holding period $t$⁠. Moreover, leveraging through margin trading accelerates changes in the futures price $dFt$ for investors, consequently elevating the probability of a margin call with higher leverage ratios.

Note that we utilize simple Brownian motion to model the probability of a margin call, consistent with Ladley et al. (2020). However, individual futures exhibit diverse characteristics, further compounded by variations across sectors, which complicates their categorization under a single stochastic process. While some futures may be accurately modeled by geometric Brownian motion, others may adhere more closely to a mean-reverting stochastic process. Given that the determination of the most appropriate stochastic process model for specific futures is beyond the scope of this study, we utilize the bootstrap simulation method to navigate this challenge. This approach allows us to incorporate variables such as the leverage ratio (initial margin), investment horizon, and maintenance margin.

There are two primary reasons for using bootstrap simulation in our study. First, bootstrap simulation allows us to comprehensively analyze the margin call risk and margin call probability under specific conditions. This approach enables us to consider potential scenarios from a vast set of outcomes within a population, utilizing historical data from futures markets. Thus, we can simulate various investment scenarios that investors might encounter in the future. As discussed by Harvey and Liu (2021) in their research on equity risk factors, there is a possibility that investors with leveraged positions may not receive a margin call purely by chance. The bootstrap simulation can address this issue by treating individual simulation trials as possible investment scenarios. By aggregating all of these potential outcomes, we can examine the probability of receiving a margin call under specific conditions. Therefore, the simulation approach has advantages in providing a practical and robust way to explore how varying leverage ratios impact leveraged returns and margin call risk.

Second, bootstrap simulation helps us overcome the limitation of sample size, particularly when conducting long-term investment research. For instance, Fama and French (2018a, b) used the monthly excess returns of the US equity market to simulate excess returns over longer investment horizons, such as 10-, 20-, and 30-year horizons. By leveraging bootstrap simulations, these studies effectively address the challenge of scarce data for longer time frames. Similarly, in our study, bootstrap simulation serves as a valuable tool to overcome the constraint of limited long-term data, allowing us to generate a sufficient number of simulated scenarios. This approach enables us to analyze the margin call risk and examine the distribution of leveraged returns across various conditions.

To understand how margin call risk is determined by margin trading via bootstrap simulation, we consider various scenarios of interest, defined by the key attributes of margin trading: the investment horizon, initial margin, and maintenance margin. We consider the investment horizon as a factor affecting margin call risk and explore various investment horizons: 1 week, 2 weeks, 1 month, 2 months, 3 months, 6 months, 1 year, and 2 years. Since we use daily excess returns, our focus is on the business days within each investment horizon (e.g. 5 business days for 1 week, 21 business days for 1 month, and 252 business days for 1 year). We also encounter differing levels of leverage (initial margin). We set the initial margin at a minimum of 7.14% of the notional value of futures contracts, which corresponds to a maximum leverage ratio of 14x. Therefore, we analyze leveraged positions by exploring scenarios in which the leverage ratio increases by 2, from 2 to 14 (or the initial margin ranges from 7.14% to 50%).^[7] For bond futures, we set the minimum initial margin at 3.33% of the notional value, allowing for a maximum leverage ratio of 30x ^[8].

Regarding the maintenance margin, we analyze three scenarios: one with no maintenance margin, one where the maintenance margin is set at 50% of the initial margin, and one where it is set at 75% of the initial margin. The absence of a maintenance margin represents a hypothetical and impractical case where margin call events occur only when investors' margin account balances drop below zero. By comparing these scenarios, we aim to evaluate the impact of varying maintenance margin requirements on futures markets. It is expected that higher maintenance margins will correspond to an increased probability of margin calls. This approach aligns with prior studies that have examined the effects of changes in margin requirements on financial markets (Daskalaki and Skiadopoulos, 2016; Heimer and Simsek, 2019). For scenarios incorporating maintenance margins, we set them at 50% and 75% of the initial margin, reflecting common market practices ^[9]. Overall, we simulate the distributions of excess returns on 60 individual futures across various investment horizons, under differing levels of initial margins and maintenance margin conditions. In total, we evaluate 10,080 scenarios (60 futures × 8 investment horizons × 7 initial margin levels × 3 maintenance margin conditions). While we analyze three maintenance margin conditions, we present results for the 50% maintenance margin in the main text and include the outcomes for the other conditions in the internet appendix for brevity ^[10].

We use bootstrap simulation techniques based on Fama and French (2018a, b); the historical sample data is used as described in the data section to represent the population of excess returns in futures markets for the bootstrap simulation. This approach essentially assumes that the actual risk premium in futures markets is equivalent to the observed risk premium during the sample period ^[11].

The simulation proceeds as follows: First, for each scenario, we randomly drew $n$ daily excess returns for the investment horizon $t$ with replacement from the full sample of daily excess returns in the data. The size of $n$ is determined by the investment horizon. For example, for a 1-week investment horizon, we draw 5 random daily excess returns from the data. Similarly, for the 1-year investment horizon, we draw 252 random daily excess returns from the data. These $n$ sampled daily excess returns are then accumulated to generate one simulated excess return for the specific investment horizon. Second, we repeat the process of drawing random daily excess returns 100,000 times with replacement to generate the simulated distribution of the leveraged excess returns for each scenario. Finally, we calculated the margin call probability based on the simulated results. A margin call occurs when the leverage loss of a simulated cumulative excess return path exceeds the maintenance margin. To determine the probability of a margin call, we define n as the number of times the simulated excess return drops below the maintenance margin level in 100,000 iterations. The probability is then calculated as n/100,000.

After generating the simulated distribution of the leveraged futures returns (considering margin call events), we examine how investors determine their optimal leverage ratios across various conditions based on the simulated distributions. This analytical approach allows us to unravel the influence of margin requirements on investor behavior and the emergence of low-risk anomalies.

We assume that investor preferences can be fully captured by the mean and variance of their chosen leverage ratios for a specific condition. When the investment horizon is denoted as $t$⁠, the decision-maker aims to select the leverage ratio $η*$ that maximizes their expected utility with leveraged returns $rl(t)$⁠. The expected utility can be expressed by Equation (2) based on the mean-variance model proposed by Markowitz (1952).

where $η*$ is the leverage ratio, $t$ is the investment horizon, $rl(t)$ is the leveraged returns for $t$⁠, and $γ$ represents the investor’s risk aversion.

To identify the optimal leverage ratio, η∗, we utilize the simulated distributions of leveraged futures returns and incorporate them into Equation (2), which enables us to consider different levels of risk aversion and investment horizons for each futures contract. We incorporate margin call risk directly into the simulated distributions of leveraged returns. When the margin call risk is high, the simulated returns exhibit lower means and greater variances compared to scenarios without margin call risk. Our approach captures margin trading and margin call risk without requiring additional assumptions or terms in the mean-variance optimization framework. This approach contrasts with Jacobs and Levy (2012, 2013), who introduced a risk aversion term to measure margin call risk.

We investigate how the presence of low-risk anomalies changes depending on the margin call risk across different sectors of futures markets. Since we define the margin call risk in terms of the investment horizon, we analyze the BAB returns in relation to different holding periods. Following Frazzini and Pedersen (2014), we construct the BAB portfolios by taking long positions in low-beta futures and short positions in high-beta futures. For equity indices, country bonds, and currencies, we compute the betas in relation to a GDP-weighted portfolio. For commodities, betas are determined in reference to a diversified portfolio that equally distributes risk across all commodities. The estimated beta for futures $i$ is calculated by Equation (3).

where $ρˆ$ is the correlation between the futures $i$ and the market returns, and $σˆi$ and $σˆm$ are the estimated volatilities of futures $i$ and the market returns, respectively.

To derive each BAB factor, we rank all futures in a sector in ascending order based on their estimated beta. Using this ranking, we divide futures into two categories: low-beta and high-beta. The futures with a beta below the sector median form the low-beta portfolio, while those above the median comprise the high-beta portfolio. Inside each portfolio, the weights of futures are set inversely to their beta values. Consequently, the low-beta portfolio gives more weight to futures with lower betas, and the high-beta portfolio favors those with higher betas.

where $rt+1Low=rt+1′wLow$⁠, $βtLow=βt′wLow$⁠, $rt+1High=rt+1′wHigh$⁠, $βtHigh=βt′wHigh$⁠, $wHigh=k(z−z¯)>0$⁠, $wLow=k(z−z¯)<0$⁠, $k=2|z−z¯|$⁠, $z=rank(βit)$ at portfolio formation, and $z¯$ is the average rank of $βit$⁠.

We perform a monthly rebalancing for these portfolios at the outset of each calendar month. For the analysis of BAB factor returns, we use the BAB returns from the original dataset in Frazzini and Pedersen (2014), spanning the period from 1965 to 2012, to eliminate any possibility that our results may be driven by measurement errors or data differences from those reported in Frazzini and Pedersen (2014) ^[12].

We analyze the BAB returns in relation to different holding periods, as we define the margin call risk in terms of the investment horizon. BAB returns are calculated using the most liquid futures contracts by cumulating daily excess returns over a one-month period, as introduced in the Data section. To construct portfolios over one-month holding periods, we use a methodology that involves overlapping the BAB returns and rebalancing on a monthly basis, following Jegadeesh and Titman (1993). If investors have different investment horizons, the observed portfolio returns can be viewed as a composition of individual investors with different investment horizons. As the holding period of the portfolio return increases, the trading positions of investors with shorter investment horizons are canceled out, highlighting the characteristics of long-term investors.

To examine how the margin call risk changes according to the holding period and conditions of the leverage ratio (initial margin) and maintenance margin, we summarize the probability of receiving a margin call based on the simulated distributions of excess returns in Table 1. We present the mean and standard deviation (SD) of the probabilities over the investment horizon and the leverage ratio for each futures sector (i.e. commodity, equity, currency, and bond). For brevity, Table 1 focuses solely on the case with a 50% maintenance margin, representing a 50% loss of the initial margin. Results for other maintenance margin conditions are provided in Internet Appendix  Table A2. These probabilities are expressed as percentage points.

We observe a consistent increase in margin call risk as the investment horizon extends across all leverage ratios for all four futures sectors, regardless of the presence of maintenance margins. These findings suggest that investors face greater pressure to increase their leverage when they have a long-term investment horizon. For example, in commodity futures with 4x leverage, the probability of receiving a margin call rises dramatically from 0.01% at a one-week horizon to 11.64% at three months, further increasing to 34.82% at one year and reaching 50.13% at two years. This pattern is similarly evident in equity futures, where at 6x leverage, the margin call probability escalates from a mere 0.05% at one week to 17.70% at three months, then to 39.12% at one year, and finally to 50.81% at two years. Even in currency futures, which generally demonstrate lower margin call probabilities, the same temporal pattern emerges. With 8x leverage, the probability increases from 0.01% at one week to 9.80% at three months, then rises substantially to 31.48% at one year and reaches 46.30% at two years. Bond futures, despite being the most stable asset class, also exhibit this pattern at 12x leverage, with probabilities increasing from nearly 0% at one week to 2.68% at three months, 10.15% at one year, and 14.16% at two years.

The data in Table 1 reveals a clear positive correlation between leverage ratios and margin call probabilities across all futures sectors. As investors increase their leverage ratios, which effectively decrease their initial margin requirements, they face progressively higher risks of receiving margin calls. This relationship becomes particularly pronounced over longer investment horizons. A notable observation from the data is the relatively minimal margin call risk during very short holding periods, even when investors employ high leverage ratios. For instance, in commodity futures, even with a substantial 14x leverage ratio, the one-week margin call probability remains at just 20.78%. Similarly, equity futures with 14x leverage show only a probability of 11.42% over the same one-week period. This pattern holds true across all futures sectors, suggesting that short-term trading strategies may accommodate higher leverage ratios without significantly increasing margin call risk.

However, the interaction between leverage ratios and investment horizons presents a compelling risk dynamic. Even modest increases in leverage ratios can lead to substantial rises in margin call probabilities as the investment horizon extends. For example, in commodity futures, raising the leverage ratio from 4x to 6x results in the margin call probability increasing from 11.64% to 32.49% over a three-month horizon. This effect becomes even more pronounced over longer periods, with the same leverage increase leading to a jump from 34.82% to 56.29% over a one-year horizon.

The analysis of the coefficient of variation (CV) across different investment horizons reveals important patterns in the distribution of margin call probabilities. The data from Table 1 demonstrates how cross-sectional differences in margin call probability decrease and converge toward the mean as the investment horizon expands. In commodity futures with a 4x leverage ratio and a one-week investment horizon, the CV for the margin call probability is 2.0 (calculated from SD of 0.02 divided by mean of 0.01), indicating substantial variation in risk across contracts in the short term.

This variation diminishes significantly over longer time horizons. For commodity futures maintaining the same 4x leverage ratio over a two-year horizon, the CV decreases to 0.26 (SD of 13.16 divided by a mean of 50.13). The convergence becomes even more pronounced with higher leverage ratios. At 14x leverage over a two-year horizon, the CV drops further to 0.042 (SD of 3.72 divided by a mean of 89.38), demonstrating that the standard deviation represents only about 4.2% of the mean probability.

This systematic reduction in CV has important implications for understanding margin call risk dynamics. As outlined in Equation (1), cross-sectional differences in margin call probabilities arise from individual futures volatility. However, the data reveals that these individual volatility effects become less influential compared to the broader impacts of leverage ratios and investment horizons as these parameters increase. This finding suggests that when investors employ higher leverage ratios or extend their investment horizons, the specific volatility characteristics of individual futures contracts become less critical in determining margin call risk. Instead, leverage and time horizon emerge as the dominant drivers of margin call probability. Figure 1 visually reinforces these findings by illustrating the convergence of margin call probabilities across all individual futures contracts over extended time horizons. This convergence is particularly evident when examining the 14x leverage scenario, where the dispersion of probabilities narrows significantly as the investment horizon lengthens.

Analysis of margin call probabilities in Internet Appendix  Table A2 shows that maintenance margins significantly amplify these probabilities. This effect is particularly evident when examining cases with a maintenance margin set at 75% of the initial margin. Under these conditions, even with a brief one-week investment horizon, commodity futures utilizing the maximum leverage ratio of 14x face a substantial mean margin call probability of 56.59%. This elevated risk level suggests that investors would likely adopt more conservative leverage positions, either by utilizing lower leverage ratios or by restricting their maximum leverage exposure to ultra-short investment periods of less than one week.

The impact of maintenance margins becomes increasingly pronounced as investment horizons extend. The data demonstrates that maintaining higher leverage ratios becomes significantly more challenging in the presence of maintenance margins compared to their absence, primarily due to the elevated risk of triggering margin calls. This relationship creates a more restrictive environment for leverage decisions, particularly over longer investment periods, effectively constraining investors' ability to maintain high-leverage positions.

Overall, our analysis provides robust empirical evidence demonstrating that margin call risk exhibits significant sensitivity to changes in investment horizons across the full spectrum of individual futures contracts. The investment horizon emerges as a crucial determinant in establishing margin call risk levels, with distinct patterns emerging across different time frames. During sufficiently short investment horizons, margin call risk maintains notably low levels, creating favorable conditions for investors to employ leverage in their trading strategies. However, as investment horizons extend, margin call risk increases substantially, potentially influencing investor preferences toward lower leverage rather than high leverage, effectively serving as a natural constraint on leverage decisions.

Given these findings, the next section examines investor decision-making processes, aiming to determine optimal leverage ratios that balance these competing factors. This analysis will provide crucial insights into how investors can optimize their leverage strategies while accounting for both investment horizons and individual risk preferences.

Table 2 presents the mean optimal leverage ratios across different futures sectors, investment horizons, and investor risk aversion rates, calculated using the mean-variance utility function specified in Equation (2). The analysis incorporates maximum leverage constraints of 30x for bond futures and 14x for other futures sectors, with results reflecting scenarios where the maintenance margin represents 50% of the initial margin.

The data reveals several significant patterns in optimal leverage ratios. First, we observe a consistent decline in mean optimal leverage ratios as investment horizons extend, with the lowest values consistently appearing at the two-year horizon. While initial optimal leverage ratios are notably high for shorter investment horizons, a pronounced decreasing trend emerges beyond the three-month mark. This pattern is particularly evident among investors with lower risk aversion rates. In contrast, investors with higher risk aversion rates (exceeding 1) generally maintain optimal leverage ratios near one across all investment horizons, with bond futures representing the primary exception to this pattern.

The empirical evidence supports a nuanced relationship between leverage utilization and investment horizons. While investors demonstrate a willingness to employ high leverage initially, they systematically reduce their leverage exposure as margin call risk escalates with longer investment horizons. For example, in commodity futures with a risk aversion of 0.10, the optimal leverage ratio decreases from 7.17 at the one-week horizon to 2.77 at the two-year horizon. Similar patterns are observed across other sectors, with equity futures showing a reduction from 11.33 to 4.89, currency futures from 8.25 to 5.25, and bond futures from 28.77 to 22.15 under the same risk aversion level.

For robustness, we examine how different maintenance margin requirements affect our results. When we increase the maintenance margin from 50% to 75% of the initial margin, our findings become even stronger. Higher maintenance margins increase the probability of margin calls, particularly for longer investment horizons, leading investors to reduce their optimal leverage more aggressively. These results reinforce our main argument that margin call risk serves as an effective leverage constraint, with its impact becoming more pronounced as investment horizons lengthen. Detailed results are presented in Internet Appendix  Table A3.

Table 3 presents the statistics of the BAB returns with respect to holding periods and sectors in futures markets ^[13]. Table 3 shows that with increasing holding periods, the statistical significance of the BAB return strengthens across all sectors. In particular, when the holding period exceeds six months, the BAB return becomes statistically significant at the 10% level across all sectors. Additionally, the BAB return is significant at nearly the 1% level for a one-year holding period across all sectors. For example, in the case of the one-year holding period, the commodity sector presents the lowest test statistic, with a t value of 2.55, which falls between the 2% significance level (t value of 2.326) and the 1% significance level (t value of 2.576). However, for the investment horizon shorter than six months, the BAB return loses significance across all sectors except for the equity sector. These results support our argument that BAB returns become insignificant in futures markets where high leverage is accessible during short-term investment horizons with low margin call risk. This suggests that low-risk anomalies, typically driven by leverage constraints and high-beta preferences, may not exist in futures markets when margin call risk is low. Again, Table 3 shows consistent results regarding the importance of margin call risk on leverage constraints in futures markets and investor decisions regarding the use of leverage.

Consistent with Frazzini and Pedersen (2014), who used a 1-month holding period, our results in Table 3 show that BAB returns are statistically significant only in the equity sector of futures markets. To understand why BAB returns remain significant only in the equity sector even in short-term investment horizons, we consider additional leverage constraints beyond margin call risk. One explanation is that equity market investors face more stringent leverage constraints, such as legal restrictions on equity funds (Frazzini and Pedersen, 2014; Boguth and Simutin, 2018). These constraints operate independently of margin call risk, potentially driving significant BAB returns in the equity sector even during short investment horizons. Moreover, as Daskalaki and Skiadopoulos (2016) show, the equity sector’s lower sensitivity to margin requirement changes suggests that the benefits of high leverage through margin trading may be weaker in equities compared to other sectors. Furthermore, the characteristics of futures market sectors differ substantially. For example, in the commodity sector, a substantial body of research focuses on identifying commodity risk premiums based on commodity-specific factors (Szymanowska et al., 2014; Bakshi et al., 2019, among others). Numerous studies have demonstrated that commodity returns often exhibit unique characteristics and lack correlation with broader market factors (Dusak, 1973; Black, 1976; Carter et al., 1983; Jagannathan, 1985; Bessembinder, 1992; De Roon et al., 2000; Erb and Harvey, 2006). In addition, unlike equity futures, there is a significant correlation between shifts in margin requirements and returns for commodity futures (Daskalaki and Skiadopoulos, 2016). There is also evidence of a prevalence of highly leveraged investors in both the foreign exchange futures and commodity futures sectors, either at the individual or account level (Heimer and Simsek, 2019; Ladley et al., 2020). Therefore, we argue that it is important to consider the significance of the BAB return within individual sectors rather than combining all futures sectors into a single portfolio.

To examine the stability of our findings across different time periods, we conduct a subperiod analysis by dividing our sample at January 2000, which represents approximately the midpoint of our bond futures data (the sector with our shortest sample period). The results, presented in Panels B and C of Table 3, show patterns consistent with our full-sample findings (Panel A). Specifically, in both subperiods, BAB returns remain insignificant across all sectors except equity for investment horizons shorter than six months while becoming increasingly significant with longer investment horizons. This consistency across different time periods strengthens our conclusion about the relationship between investment horizons, margin call risk, and BAB returns in futures markets.

In conclusion, our analyses show that low-risk anomalies resulting from leverage constraints in futures markets become more pronounced as the margin call risk becomes more significant over longer investment horizons. However, when the holding period is short (six months or less, according to our results), investors do not need to consider low-risk anomalies in commodity, currency, and bond futures markets.

One might argue that factors other than margin call risk could explain why BAB returns increase with investment horizons. For example, a natural consideration would be the role of volatility. If investors avoid assets above certain volatility thresholds, longer investment horizons (which increase asset volatility by $horizon$ × volatility) could lead to greater aversion of high-volatility assets. However, this would predict a lower preference for high-beta assets over longer horizons, contradicting our findings. Alternatively, if investors become less sensitive to volatility over longer horizons, potentially developing stronger high-beta preferences, this could explain strengthening BAB returns. Yet, this alternative explanation cannot fully account for our findings, particularly the insignificant BAB returns in short-term futures markets. Moreover, even if such volatility preferences exist, they do not negate the role of margin call risk as a leverage constraint since long-term investors with high-beta preferences must still manage their leverage to avoid margin calls. We leave the investigation of additional channels for future research.

The holding period of individual investors in futures markets is relatively short (Subrahmanyam et al., 2024). Although evidence on the actual holding period of individual investors is scarce due to data limitations, Subrahmanyam et al. (2024) provide empirical evidence that the holding period of most individual investors in futures markets is relatively short. The study found that 99% of individual investor holding periods are less than 34.42 h; these results are based on the account-level transaction data of 10,507 individual investors involving 1,086 contracts written on 51 different underlying assets in the Chinese futures markets from 2014 to 2016. Therefore, our results indicate that, on average, most individual investors are not significantly affected by low-risk anomalies in commodity, currency, and bond futures markets.

On the other hand, investors with longer holding periods, such as hedge funds, pension funds, and sovereign wealth funds, may consider low-risk anomalies in futures markets. For example, hedge funds and commodity trading advisors have used futures contracts for managed futures strategies since at least the 1970s (Hurst et al., 2013). Pension funds and sovereign wealth funds utilize futures markets due to the benefits of diversification from an asset allocation perspective for their long-term investment horizons (Jen, 2010; Al-Hassan et al., 2018; OECD, 2022). Numerous studies show that investing in futures markets adds value to a diversified portfolio (Edwards and Park, 1996; Jensen et al., 2000; Gorton and Rouwenhorst, 2006). Additionally, Levine et al. (2018) show that commodity futures provide a hedge against positive inflation shocks, unlike stocks and bonds, in the long run.

Furthermore, institutional investors, who tend to have longer-term investment horizons than individual investors, hold the largest portion of positions in futures markets. Internet Appendix  Table A4 shows the percentage of open interest by trader category in financial futures positions, based on Commodity Futures Trading Commission data for the week ending June 11, 2024. These categories include dealer/intermediary, asset manager/institutional, leveraged funds, other reportables, and nonreportable traders. The “Asset Manager/Institutional” category, which comprises institutional investors such as pension funds, endowments, insurance companies, mutual funds, and portfolio/investment managers whose clients are primarily institutional, holds the largest portion of the outstanding long and short positions. Hedge funds that use managed futures strategies are classified in the “Leverage Funds” category. In contrast, individual investors, classified as nonreportable traders, account for an average of only 10% of open interest for both long and short positions in U.S. financial futures. Overall, low-risk anomalies in futures markets attributable to margin call risk can be viewed as substantial.

The availability of margin trading, which allows investors to leverage their positions up to 10–30 times, relaxes leverage constraints in futures markets compared to other financial markets. However, evidence regarding the extent of leverage constraints in futures markets is scant and even inconsistent. In this study, we suggest that the margin call risk should be considered when identifying leverage constraints in the futures market.

Our findings suggest that the presence of low-risk anomalies in futures markets is influenced by the level of margin call risk, which is proportional to the investment horizon and acts as a leverage constraint in futures markets. We find that the investment horizon plays an important role in determining the probability of receiving margin calls and, consequently, investor decisions to use leverage. In particular, longer investment horizons increase the margin call risk, which puts investors under greater pressure to decrease leverage over longer investment horizons. Therefore, as the investment period lengthens, an investor’s optimal leverage ratio decreases, which means that although investors tend to use high leverage, they reduce their leverage levels as their investment horizons lengthen in response to increased margin call risk.

Interestingly, over a very short holding period, the probability of a margin call risk occurring is minimal even at low initial margins (i.e. high leverage ratios). Our results show that when the investment horizons are sufficiently short, the margin call risk remains low, making it relatively easy for investors to use leverage. We also find that investors tend to use high leverage when the margin call risk is low. Moreover, from the BAB analyses, low-risk anomalies are not a significant systematic risk factor in futures markets with relatively short holding periods, further implying that investors are not leverage-constrained in futures markets over short investment horizons.

Our study is similar to previous studies that provide evidence of the existence of leverage constraints in financial markets but differs from them in that we consider the margin call risk, a feature that distinguishes leverage constraints in futures markets from those in other financial markets. Additionally, unlike previous studies that required the assumption of a mean-variance optimization framework to capture margin trading and the margin call risk, we directly incorporate the margin call risk into the simulated distributions of leveraged returns and examine the investor decision-making processes. Our methodology simplifies the understanding of margin trading by providing a simple and direct means of accounting for the risks inherent in margin calls in futures trading.

In conclusion, we provide empirical evidence that the features of margin trading affect the presence of low-risk anomalies in futures markets. While easy access to leverage through margin trading may suppress the occurrence of low-risk anomalies, the emergence of leverage constraints due to an increased margin call risk can lead to the presence of low-risk anomalies. Therefore, in futures markets, margin trading may act as a constraint on the use of leverage in situations in which the margin call risk is high, but it also alleviates investor leverage constraints in situations in which the margin call risk is low.

The supplementary material for this article can be found online.