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As mentioned several times, Quantpedia is a big fan of transferring ideas from one asset class to another. This article is another example; we use an idea originally tested on Chinese stocks and apply it to the commodity futures investment universe. The resultant return new asymmetry investment factor in commodities is an interesting trading strategy unrelated to other common factors and has a slightly negative correlation to the equity market and can be therefore used as an excellent diversifier in multi-asset multi-strategy portfolios.

Introduction

A probability distribution is a function that describes all the possible values and likelihoods that a random variable can take within a given range. Several factors determine where the value is likely to be plotted, including the distribution’s mean, standard deviation, skewness, and kurtosis. Academics and investors use probability distribution to evaluate the possible expected returns that the asset may yield in the future.

Theoretical studies show that the greater (lower) upside asymmetry is associated with lower (higher) expected returns. Consistent with the theory, Lin and Liu (2017) found that individual investors are willing to pay for a small probability to win a significant payoff or, in other words, to buy stocks with a high skewness. According to the authors, the aforementioned leads to a negative relationship between skewness and return in the cross-section. Perez et al. (2017) were the first to examine the skewness in the commodity futures. Specifically, their trading strategy that takes long positions in commodity futures with the most negative skewness and shorts those with the most positive skewness generates a significant excess return that remains statistically significant after controlling the exposure to well-known risk factors.

Padyšák and Vojtko (2019) studied the performance of the commodity skewness portfolio during bear markets. In particular, their portfolio went long on the bottom four commodities with the lowest skewness while shorting the top four with the highest skewness. They found that when the monthly performance of the S&P 500 index was negative with an average monthly return of -3.49%, the average monthly return of the commodity skewness strategy was 1.10%.

Wu, Zhu, and Chen (2020) examined return asymmetry in the Chinese stock market with a new distribution-based asymmetry measure (IE) of Jiang et al. (2020). They sorted the Chinese stocks into deciles and found that the average value-weighted return spread between the highest and lowest IE decile is -0.81% (t-value = -2.70) per month, which is statistically significant at the 1% level.

This paper aims to examine the return asymmetry in the commodity futures in a slightly less traditional way. Instead of using skewness as a proxy for the return asymmetry, we rely on a new asymmetric measure IE proposed by Jiang et al. (2020), that uses the difference between upside and downside return probabilities to capture the degree of asymmetry. The greater the measure, the greater the upside potential of the asset return. Typical risk-averse investors prefer extreme gains and avoid extreme losses. Consequently, they bid up the prices of assets with a high chance of extreme gains and pay a lower price for assets with a high likelihood of extreme losses. As a result, the high (low) IE assets become overvalued (undervalued), and their subsequent returns are lower (higher). Moreover, the implementation is truly simple as the theoretical definition of the asymmetry measure leads to a metric that is equal to the past number of days with extremely high minus extremely low returns divided by a number of days in the sample.

As for our investment universe, we follow Padyšák and Vojtko (2019) as their dataset of commodities offers broad exposure to the commodity market. We construct our eleven long-short portfolios based on the asymmetric measure as follows. At the beginning of each month, we calculate asymmetry (IE) for each commodity using the latest 260 daily returns. Then we rank commodities according to their IE. The portfolios go long on the bottom m commodities with the lowest IE in the previous month and short on the top m commodities with the highest IE in the previous month.

While various possibilities are studied, a portfolio that is associated with m equal to seven exhibited the highest return, greatest t-stat, and the largest Sharpe ratio among all examined portfolios. The portfolio goes long on the bottom seven commodities with the lowest IE in the previous month and shorts the top seven commodities with the highest IE in the previous month. It achieved a 0.38% mean monthly return, which is statistically significant at a 1% level (t-statistic = 3.33). In other words, with a 4.36% annual return, a dollar invested in 1991 would be worth 3.64 dollars in 2021. From a risk-adjusted perspective, it exhibits a risk-adjusted return of 0.58. Additionally, following Padyšák and Vojtko (2019), we examined the performance of “Portfolio 7” during the periods when the S&P 500 monthly return was negative. We found that when the monthly S&P 500 return was negative, the average monthly S&P 500 return was -3.50%, while the average monthly return of the “Portfolio 7” was 0.50%. We explained these findings with the negative correlation between the returns of the examined time series. Specifically, we regressed “Portfolio 7” returns on the S&P 500 returns and found a negative slope coefficient of β = -0.04 (t-stat = -1.53). The correlation between the representative IE portfolio and S&P 500 returns varied over time with an average correlation coefficient of ρ = -0.08 (t-stat = -1.53). Therefore, it seems that the systematic commodity strategy based on asymmetry could be utilized during stock bear markets. By implementing the proposed strategy into the stock portfolio, the investor can partially cover the losses during stock market downturns.

A reasonable question arises, why not simply rely on the well-known skewness effect during times of market turmoil. With its increasing popularity, every strategy is becoming less profitable. McLean and Pontiff (2015) conclude that investors are aware of academic publications and learn about mispricing. As a result, there is a decay in the performance of anomalies and we can expect that it also does hold for the skewness effect. Additionally, the correlation analysis shows that although the skewness and asymmetry effects are related, the correlation is not that high, and both effects form distinct trading strategies. Therefore, each novel, unique, and statistically significant anomaly could be interesting in the investing/trading practice.

Data

In our study, we use data starting in April 1991 and ending in July 2021. We obtained the daily commodity futures returns from Quandl and the SPY monthly returns from Yahoo Finance. Our sample consists of 22 commodity futures, namely: soybean oil, corn, cocoa, cotton, feeder cattle, gold, copper, heating oil, coffee, live cattle, lean hogs, natural gas, oats, orange juice, palladium, platinum, soybean, sugar, silver, soybean meal, wheat, and crude oil.

Methodology

We start building the eleven portfolio sorts as follows. At the beginning of each month, we calculate asymmetry (IE) for each commodity. We assess the upside asymmetry by calculating its excess tail probability as proposed by Jiang et al. (2020):

where f(x) denotes the density of commodity returns and c_{1} is equal to the two standard deviations plus the mean and c_{2} is equal to the mean minus two standard deviations. The first argument measures the cumulative probability of extreme gains and the second argument the cumulative probability of extreme losses. The positive (negative) IE indicates that the probability of the extreme gains overweights (underweights) the probability of extreme losses. Investors seem to prefer extreme gains and avoid extreme losses. Consequently, they bid up the prices of assets with a high chance of extreme gains and pay a lower price for assets with a high likelihood of extreme losses. As a result, the high (low) IE assets become overvalued (undervalued), and the subsequent returns are lower (higher).

Naturally, we do not exactly know the density of commodity returns in equation 1 and we need to use the empirical density (or empirical distribution function) based on the past 260 trading days or approximately one year. Since the integrals represent the probabilities of extremely positive (negative) returns, we compute the IE as a difference between the probability of extreme gains P_{j} (X > c_{1}) and the probability of extreme losses P_{j} (X < c_{2}). Using empirical estimates, equation 1 significantly simplifies and the IE can be computed as follows:

where r_{ij} denotes the return of commodity j on the day i. The first sum represents the number of trading days when daily return r_{ij} is greater than c_{1}, and the second sum the number of trading days when daily return r_{ij} is smaller than c_{2}. Then, we construct a signal for the commodity j as: S_{j} = rank (IE_{j}). Finally, we can compute the return of the Portfolio m in the subsequent month r_{t+1}_{ }as:

where m = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. The first sum denotes that our portfolio goes long on the bottom m commodities with the lowest IE in the previous month, and the second sum that we go short on the top m commodities with the highest IE in the previous month. Therefore, the portfolio is equally weighted and rebalanced monthly.

Empirical Results

Table 1 presents the performance results of our eleven portfolios. The monthly returns of the portfolios are computed following formula (3) and then geometrically annualized. The Portfolio m goes long on the bottom m commodities with the lowest IE in the previous month and shorts the top m commodities with the highest IE in the previous month. For example, Portfolio 7 goes long on the bottom seven commodities with the lowest IE in the previous month and shorts the top seven commodities with the highest IE in the previous month. The risk-adjusted return is calculated as the annualized return divided by the annualized volatility. Statistical significance at the 1%, 5%, and 10% levels is indicated by ***, **, and *, respectively. Best values are in bold.

The results in the table indicate that including the additional commodities in the portfolio lowers the portfolio volatility and maximal drawdown. Specifically, Portfolio 1 has the highest volatility with the most severe maximal drawdown, while Portfolio 11 has the lowest volatility and the lowest maximal drawdown. The relationship between the number of commodities in a portfolio and the respective return is not linear. The returns rise up to Portfolio 7 and then gradually decline till Portfolio 11. Overall, almost all portfolios have statistically significant and economically large returns. Specifically, the returns of the Portfolios 2-11 are statistically significant at least at the 5% level. We can observe a similar pattern when looking at the risk-adjusted returns. Portfolio 7, which has the highest return, exhibits the largest risk-adjusted return. The relationship between our portfolios and their respective risk-adjusted returns shows the Figure 1.

So far, we have briefly reviewed the portfolio sorts. Now we will study Portfolio 7 in detail, as it exhibits the highest return and greatest t-stat.

Portfolio 7 goes long on the bottom seven commodities with the lowest IE in the previous month and shorts the top seven commodities with the highest IE in the previous month. It achieved a 0.38% mean monthly return, which is statistically significant at a 1% level (t-statistic = 3.33). In other words, with a 4.36% annual return, a dollar invested in 1991 would be worth 3.64 dollars in 2021 as shows the Figure 2.

From a risk perspective, the maximal drawdown was of the Portfolio 7 was -29.03%. The major downturn lasted for 37 months, while the recovery took almost 2 years. The visualization of the drawdowns shows the Figure 3.

Furthermore, following Padyšák and Vojtko (2019), from 1994 to 2020, we examined the performance of Portfolio 7 during the periods when the S&P 500 monthly return was negative. There were overall 324 monthly observations of which 119 with a negative S&P 500 monthly return. We found that when the monthly S&P 500 return was negative, the average monthly S&P 500 return was -3.50%, while the average monthly return of the Portfolio 7 was 0.50%. We can explain these findings with the negative correlation between the returns of the examined time series. Specifically, we regressed the Portfolio 7 returns on the S&P 500 returns and found the negative slope coefficient of β = -0.04 (t-stat = -1.53). The correlation between Portfolio 7 and S&P 500 returns varies over time with an average correlation coefficient of ρ = -0.08 (t-stat = -1.53). Figure 4 shows that the negative relationship becomes more significant during stock market declines like in the early 2000s, 2007, and 2020. Therefore, we propose to use our strategy as a possible (partial) hedge to the stock portfolio. By implementing the proposed strategy into the stock portfolio, the investor can partially cover the losses during stock market downturns.

Since skewness and asymmetry effects are based on a similar idea, it is crucial to examine their relationship. Therefore, we have examined whether the skewness effect does subsume the returns of our strategy. Specifically using the same investment universe, we have created another eleven portfolio sorts based on the skewness effect. Portfolio Skew s goes long on the bottom s commodities with the lowest skewness in the previous month and shorts the top s commodities with the highest skewness in the previous month where s = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. For example, Skew 2 goes long on the bottom 2 commodities with the lowest skewness in the previous month and shorts the top 2 commodities with the highest skewness in the previous month.

Conclusion

To sum up the results, we have expanded the new distribution-based asymmetry measure (IE) of Jiang et al. (2020) into the new asset class. Consistent with the previous literature, assets with the largest probabilities of extremely high returns underperform, while assets with high probabilities of extremely low returns outperform. Additionally, the theoretical definition of the asymmetry measure leads to a metric that can be easily computed when one uses the empirical distribution. In the practice, the measure is equal to the past number of days with extremely high minus extremely low returns divided by the number of days in the sample.

We have constructed eleven long-short commodity portfolios and studied the best-performing portfolio in detail. We proposed a strategy that goes long on the bottom seven commodities with the lowest IE in the previous month and shorts the top seven commodities with the highest IE in the previous month. It achieved a 0.38% mean monthly return, which is statistically significant at a 1% level (t-statistic = 3.33). In other words, with a 4.36% annual return, a dollar invested in 1991 would be worth 3.64 dollars in 2021. Furthermore, we examined the performance of our strategy during the periods when the S&P 500 monthly return was negative. We found that when the monthly S&P 500 return was negative, the average monthly S&P 500 return was -3.50%, while the average monthly return of our strategy was 0.50%. We explained these findings with the negative correlation between the returns of the examined time series. Additionally, the correlation analysis shows that although the skewness and asymmetry effects are related, the correlation is not that high, and both effects form distinct trading strategies.

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