Mexican White Corn Spot Price Hedging with US Agricultural Futures Portfolios Using the Surplus Efficient Frontier

Abstract

This paper addresses the lack of hedging effectiveness that yellow corn 1-month futures of the Chicago Mercantile Exchange (CME) offer for cross-hedging the price of Mexican white corn. For this purpose, the authors tested 1013 combinations (portfolios) of the ten most traded futures on the CME and the New York Mercantile Exchange (NYMEX). The results suggest that using a 51.6741% corn and a 48.3259% wheat portfolio mimics the white corn price with a hedging effectiveness of 0.6180. To test the practical use of such a portfolio, the authors backtested its use from 1 January 2000 to 9 February 2025 as a balancing short position for sale of white corn at t + 1. By using the corn–wheat portfolio, the simulated seller (farmer or intermediary) would have earned MXN 5.7664 per kilo traded. The results in this paper provide the first solution to the Mexican white corn cross-hedging problem with a futures portfolio. This hedge can be used as the balancing (short) position for the strike or minimum buy price that the Mexican Government or a financial institution could offer to farmers and intermediaries to enhance food security.

1. Introduction

Agricultural production is an activity subject to several risks that impact the performance and livelihood of producers and some related intermediaries. This situation is particularly important for medium- and small-sized producers, who depend on favorable prices and weather conditions for their profitability. In this paper, the authors propose a method to hedge income risk for Mexican white corn (corn, henceforth) sellers (producers or intermediaries). Managing this type of risk is particularly relevant to enhancing the country’s corn supply. Although it is among the leading corn producers worldwide, Mexico is a highly dependent country in terms of corn imports, due to the reduction in production levels in recent decades and the growing internal demand. Corn is the main staple of the Mexican diet, and its price could impact the consumption and welfare of all income levels. Consequently, the proper formation and stability of corn prices are essential goals for the Mexican Government.

Corn is a staple that depends on weather conditions and fluctuations in market prices. Low prices mean lower income and, consequently, a lower supply of corn (small and midsize producers are affected by these fluctuations). High prices affect the benefits of intermediaries as well as household consumption and overall inflation levels. Mexico is among the top ten corn-producing countries, according to the Food and Agriculture Organization’s (FAO) 2023 figures [1]. Figure 1 illustrates the historical harvested area by geographical region and worldwide, providing an initial overview of corn production. As noted in the figure, Eastern Asia, North America, and South America are among the leading areas. Of these, the top ten producing countries are the United States, China, Brazil, India, Argentina, Mexico, Nigeria, Ukraine, the United Republic of Tanzania, and the Democratic Republic of Congo. Figure 2 displays the historical production (in hectares or ha) of these countries.

As noted in Figure 2, Mexico’s corn production has decreased from 5.95% of the global output in 1961 to 3.09% in 2023. This is a result of the impact that lower prices had on producer income during the 1970s and 1980s [2,3,4]. Given the lower prices, some small and mid-sized producers turn to another crop or close their plantations. Only those who had economies of scale by increasing their harvested area or had the capital and capacity to adopt new technologies remained. This situation created a competitive gap between them and the remaining medium-sized and small producers. Figure 3 shows the historical comparison of crop yields between global production and that in Mexico, as well as the countries that produce more than Mexico. The number of kilograms per hectare (ha) in the US, Argentina, China, and Brazil and the weighted regional average of the global crop yield have historically been higher than the Mexican one. This figure explains one perspective of the dependence of corn imports that Mexico has on other countries like the US (its leading commercial partner and neighbor).

Complementary to this situation, white corn production in Mexico is an essential economic activity. Following Figure 4, the general value of this grain’s production contributes about 3.0% of the gross domestic product (GDP) on average.

Following the decline in white corn supply in the 1970s, Mexico faced a shortage of white corn that threatened general consumption, particularly among lower-income populations. To mitigate such an impact on general food security, the Mexican government implemented floor or minimum price (warranty or strike price) policies. It began purchasing white corn crops directly from producers. This corn was stored in public silos and sold to the low-income population at lower prices through Public (Government) stores known as CONASUPO (the acronym in Spanish for National Popular Subsistence Company). The difference (income loss) between the minimum or strike price paid ($K$) to the producer and the lower sale price was paid with tax contributions through CONASUPO. With this mechanism, the Mexican Government promoted food security in terms of corn consumption and other staples, such as beans and white milk. In the 1990s, this organization was under public scrutiny due to corruption and was later dissolved in the 2000s. Two organizations, known as PROCAMPO (Direct Agricultural Support Program) and ASERCA (Agricultural Markets Development and Commerce Agency in Spanish), replaced the CONASUPO strike or minimum price program with direct payments to compensate beneficiaries for income loss, provided they followed specific educational and household goals, such as having their children attend school.

These three programs had an income insurance scheme by offering a minimum buy (strike) price published on a given date. This was a response by the Mexican Government (during its multiple administrations from the 1990s to 2016) to protect food security by safeguarding the supply side of several agricultural products, such as corn.

In 2016, the Mexican Government implemented a new program known as SEGALMEX (Mexican Food Security Agency), a financial agency that is part of the Secretary of Agriculture [5]. Its main goal was to implement, once again, a hedge or income insurance scheme by publishing an official ask (buy) price, estimated with the monthly mean strike price of the Chicago Mercantile Exchange (CME) 1-month yellow-3 corn futures and the mean close value of the Mexican peso/US dollar (MXNUSD) exchange rate in that period. This scheme was again funded with taxpayers’ contributions and represents a fiscal burden to the Mexican Government.

The intended effort to hedge a non-commodity staple with a traded future is known as cross-hedging. As an example, it involves using the future of an agricultural commodity to hedge a non-commodity, such as white corn. US yellow corn is a commodity with high production and trading volumes globally, unlike varieties such as Mexican white corn, popcorn, or other staples like beans or avocados in Mexico (for example). The problem with cross-hedging practices is the presence of basis or basis risk. The basis is the difference between the futures price change ($\Delta F_t$) and the spot (non-commodity) price. The primary rationale for an appropriate hedging with derivatives is $\Delta P_t-\Delta F_{i,t}\approx 0$. The basis is when $\left|\Delta P_t-\Delta F_{i,t}\right|>0$. Basis risk is defined as ${\sigma }^2\left(\Delta P_t-\Delta F_{i,t}\right)>0$. The presence of basis risk is not suitable for hedging activities and affects the income risk for corn producers. Related to this issue in the white Mexican corn market cross-hedging with commodity futures, the work of Ortiz-Arango and Montiel-Guzman [6] is the first to test the cointegration and hedging effectiveness of the CME yellow corn future in the Mexican white corn price of the leading corn-producing states in Mexico (Sinaloa, Jalisco, Michoacán, the state of Mexico, Guanajuato, Chihuahua and Guerrero). The authors found that only the State of Michoacan showed a significant relationship (in the short and long term) between its price and the corn futures price. They use the Engle–Granger [7] cointegration test to determine the presence of a long-term (cointegrating) relationship and stochastic volatility models (like multivariate GARCH models) to estimate the significance of the time-varying and dynamic correlation (the short-term relationship test). Building on this work and other studies reviewed in the following section, this paper extends the work of Ortiz-Arango and Montiel-Guzman, testing a potential solution to mitigate basis risk and enhance hedging effectiveness.

Hedging effectiveness is a measure proposed primarily by Ederington [8] and is based on the futures hedging theory of Working [9,10,11]. This measure is a sort of R-squared estimation between two positions: (1) a hedged spot price position, $\Delta {\%P}_t-\Delta \%F_{i,t}$, and (2) an unhedged one, $\Delta {P\%}_t$. The hedging effectiveness, $H{I}_{i,t}$, is defined as

$$H{E}_{i,t}=1-{\displaystyle \frac{{\sigma }^2({\Delta \%P}_t-{\Delta \%F}_t)}{{\sigma }^2\left({\Delta \%P}_t\right)}}$$

(1)

In the previous expression, the basis risk, ${\sigma }^2\left(\Delta {\%P}_t-\Delta {\%F}_{i,t}\right)$, was defined as the variance in the spot price change minus the short or opposite position in futures (change in the hedging future price). Also, in that expression, ${\sigma }^2\left(P_t\right)$ is the unhedged position variance. Hedging effectiveness (1) is a straightforward metric to determine how the simulated futures portfolio (or a single future position) fits the spot price of interest. To have a reference of an appropriate hedging (or price replicating) level, a proper hedge is when $H{E}_{i,t}\approx 1$, that is, when the basis risk is close to zero (${\sigma }^2\left(\Delta {\%P}_t-\Delta {\%F}_{i,t}\right)\approx 0$). An acceptable range of hedging effectiveness is $0.5<H{E}_{i,t}\le 1$. If $H{E}_{i,t}<0$, this means that the futures’ position increases basis risk.

To date, it has been found that the CME yellow corn future is not an appropriate hedge for Mexican white corn prices, even if SEGALMEX proposed an interesting minimum price (income risk hedge) scheme. The authors of this paper propose a potential solution: instead of using a single yellow corn futures position like SEGALMEX, use a portfolio of agricultural futures traded on either the CME or the New York Mercantile Exchange (NYMEX). The goal of the current research is to test the performance of this portfolio in mimicking the Mexican white corn price over time.

The primary motivation of the present test is to select an optimal futures portfolio $FP=w \ast =\left[w_f\right]$ whose percentage variation $\Delta \%FP$ replicates or, at least, approaches that of the Mexican white corn spot price in Mexico. The working hypothesis tested herein is that “using an agricultural futures portfolio leads to a hedging effectiveness of $H{E}_{i,t}\approx 1$ and reduces basis risk significantly”.

For this purpose, this paper will assume that a financial institution can offer a hedging price, $K$, at t + 1 week. To reduce the impact on the institution’s shareholders (if it is a private institution) or taxpayers (if it is a public one like SEGALMEX or FIRA—Trust Funds for Rural Development). The practical use of the mimicking portfolio is that these institutions can buy an opposite (short) position in it to offset the spot price difference $K_t-S_t$. This paper will test whether the selected optimal portfolio leads to a proper replication of white corn’s price.

Consequently, the working hypothesis to test in this paper is the following:

H₀: 

The use of an agricultural futures portfolio selected with the surplus efficient frontier method leads to high hedging effectiveness for the replication of the Mexican white corn price.

It is essential to highlight that this paper tests a cross-hedging method because even if there is a yellow corn future traded in the CME, the specific Mexican corn is a white species. The specific (local) market could show some diverging factors that impact the spot price, making the stochastic process of such a price different to that of the CME future (as Ortiz-Arango and Montiel-Guzman [6] found). Also, the use of an agricultural futures portfolio makes this practice a cross-hedging one because the hedging method is performed with a future (portfolio) different from the spot price of interest. As noted in the following section, there are also cointegration hedging methods that could be potentially used, but they are limited by the existence of a long-term relationship between the futures portfolio and the price of the Mexican white corn. This is an assumption that does not hold, as the Methodology Section shows.

Given this paper’s main motivation, its structure is as follows: The next section presents a literature review of studies related to hedging effectiveness, hedge ratios, and, most importantly, those that have tested the hedging effectiveness of cross-hedging practices in agricultural products worldwide. This section discusses the selection of the surplus efficient frontier method and how it works. The third section outlines the data gathering and processing steps, along with the replication results for the white corn price. In this section, the authors present a test of a quantitative hedging rule, demonstrating that using the optimal portfolio leads not only to proper price replication but also to a good hedge and reduced income risk. Finally, the last section presents the main conclusions and guidelines for further research.

2. Literature Review

The Introduction Section depicted a brief evolution of the food security mechanisms implemented in Mexico. In nearly all schemes, the Mexican Government offered a strike or minimum buy price for Mexican corn (among other staples). The difference between such a strike price and the spot price at which the corn (in the case of CONASUPO) or the claimed payment was exercised was paid by taxpayers. The World Trade Organization [12,13,14] has a food security practices (subsidy programs) classification that organizes almost all the practices of the country members in three groups or colors:

• Amber box practices: The practice of subsidizing production quantities or taking measures to support prices, such as strikes, minimum price fixing, or subsidizing income insurance or derivatives, is among these. Mexico and the United States are two countries in this group because these two governments either pay the minimum price directly to the producer (Mexico) or subsidize the payment of income risk insurance or derivative risk premiums (such as those paid in hedging activities with options). This group of practices is defined in Article 6 of the WTO Agreement on Agriculture [12].
• Blue box practices: This group of practices includes those of the amber group, as well as the use of any farmer production limitation policies. This group of practices is rarely observed among WTO members and is defined in the fifth paragraph of Article 6 of the WTO’s Agreement on Agriculture.
• Green box practices: This is the ideal group, as defined by WTO standards. In this set of practices, governments can subsidize farmers’ income without distorting markets (unlike amber and blue practices). An example of this is the European Union (EU) Income Stabilization Tool (IST), which provides subsidies to agricultural unions in EU member states to establish mutual funds and, if necessary, make contributions to such funds. The goal of this mutual fund is to pay a subsidy if the farmer’s income is below 70% of the Olympic mean of the last three years’ income.

The primary objective of this paper is to demonstrate that it is feasible to replicate the Mexican white corn price using a portfolio of agricultural futures (henceforth referred to as a mimicking portfolio), illustrating the potential to hedge the corn price with a short position and to reduce the hedging burden to taxpayers that the current SEGALMEX scheme has. Also, one of the main implications of this paper’s results is the feasibility of turning the actual WTO amber practice of a minimum price scheme into a green one. Because the government could only buy the short position in the replicating portfolio, the loss between the spot and this minimum price is no longer subsidized by taxpayers. It could be transferred to the futures market. Consequently, the idea of having a proper replication (high hedging effectiveness) of the white corn price with such a portfolio is a fundamental result to test herein.

The core idea is to translate the risk of the offered hedge from taxpayers to derivatives markets. Consequently, the Mexican Government would not be involved in subsidizing the potential loss by farmers, thereby releasing these resources for other economic policy matters.

For the Mexican case, two previous works tested the practice of cross-hedging in agricultural products. The first work is that of Ortiz-Arango and Montiel-Guzmán, which shows that there is no statistically significant short- or long-term relationship between the CME yellow corn futures price and the Mexican white corn spot price. The second is that of De la Torre-Torres et al. [15] who conducted a test similar to the one presented in this paper. The authors tested the replication of the Mexican Hass avocado (a non-commodity staple in the Mexican diet) using a portfolio of agricultural futures traded on the CME and NYMEX. The authors found that using a portfolio of 83.48% in sugar and 16.52% in coffee leads to a hedging effectiveness of 0.94 for avocado prices. This result suggests that it is feasible to use such a portfolio to offer a strike or minimum buy price for Hass avocados with a short position in the futures (sugar and coffee) portfolio. This paper aims to extend the results of these two papers to test whether it is feasible to have proper hedging effectiveness and price replication of the white corn price in Mexico.

To determine if a futures position is appropriate for hedging the price of a staple (commodity or not), there are two measures to judge such a hedge. One is hedging effectiveness $H{E}_{i,t}$, as defined in (1), and the other closely related concept is the hedge ratio, or the number of future contracts that the hedger must hold to mitigate basis risk. For this purpose, several hedge ratios have been proposed, as summarized by Myers [16]. From all these, the authors of this paper used Ederington’s [8] functional form using the percentage price variation in the spot and futures position:

$$\Delta \%P_t=\alpha +\beta \Delta \%F_t+{\epsilon }_t$$

(2)

The other functional forms, which are more complete in terms of the information set that models the future spot position relationships (such as the generalized autoregressive one of Myers), are set aside because the percentage price variation takes a uniform scale or unit. To give a general idea, the corn future is quoted in dollar cents per bushel, as is the case with soybean or wheat. Despite this, the difference between these bushels is that the corn bushel equals 25.4 Kg and soybean 27.2 Kg. Similarly, the coffee and sugar futures are quoted in pounds. Complementary to this issue, the number of units to which each futures contract is standardized differs, leading to the use of the percentage price or future variation to mitigate the impact of these different quotations. Also, using percentage price variations reduces the effect of scale in the proper price variation, $\Delta P_t,\Delta F_t$, in high price levels, leading to a reduction in the presence of heteroskedasticity in (2).

One of the first works to suggest not using a single future for cross-hedging is that of Elam, Miller, and Holder [17], who tested potential agricultural futures to hedge the price of rice bran. These authors tested the use of corn, oats, soybean, and wheat futures to hedge the monthly Arkansas rice bran price from January 1972 to December 1982 and tested either the single or combined (a sort of portfolio) futures position. Elam et al. used the linear regression coefficients of the future as the weight ($w_i={\beta }_i$) of the futures portfolio.

$${\Delta \%P}_t=\alpha +\sum _{i=1}^n{\Delta \%\beta }_{i,t}+{\epsilon }_t$$

(3)

This work is also a primary influence on this paper because it concludes that corn futures are the best hedging option for rice bran prices and suggests, as a potential solution, using a combination or portfolio of futures to hedge (replicate) the price.

One of the reasons for adopting cross-hedging with a future or a portfolio of futures is that not all staples that qualify for a commodity classification possess the necessary trading qualities. Sanders and Manfredo [18] showed that the lack of a proper contract design, the lack of industry buy-in, and the insufficient education of final users were critical factors that led to the failure of the Minneapolis Grain Exchange white shrimp future contract. To avoid such a complication, a futures portfolio used to replicate the price of the staple of interest could be a potential solution.

Related to Elam, Miller, and Holder [17] and Sanders and Manfredo [18], the latter authors developed a testing method to determine whether a given future contract’s price, $F_{i,t}$, is preferable to another, $F_{j,t}$. For this purpose, they suggested using the hedging effectiveness, determined by the residuals of each future with the hedged spot price ($P_t$). This led the authors to establish the following auxiliary regression for each pair of tested futures, given their hedge ratio equation in (2):

$${\epsilon }_{i,t}=\alpha +\lambda \left({\epsilon }_{i,t}-{\epsilon }_{j,t}\right)+{\nu }_t$$

(4)

For these authors, a given future contract, $F_{i,t}$, is preferable to another, $F_{j,t}$, if $\lambda >1$ in (4). If $0\le \lambda \le 1$, then the hedger could use a linear combination of both futures in a quantity provided by $\lambda$.

This paper is essential for the present paper because it suggests that the future-specific weight of each future in the hedging futures portfolio must add up to 1 or 100% (as it is in a conventional portfolio, where the budget restriction is given by $\sum w_i=1$). Another influence of this work is the fact that the future’s weight in the portfolio is determined by the magnitude of the residuals (a rationale that leads to the concept of tracking error or surplus frontier herein). The lower the residuals, the more dominant or preferable the future will be. This led the authors to select the surplus efficient frontier portfolio selection method, as primarily suggested by Sharpe and Tint [19] and later extended by Waring [20,21].

Following these works, it is necessary to mention the practice of passive portfolio management, which involves replicating the performance of a given benchmark or reference variable. The solution proposed in this paper to address the lack of hedging effectiveness in potential cross-hedging practices for Mexican white corn is to use a futures portfolio to replicate its price. This portfolio must be selected to minimize basis risk, ideally approaching zero. That is, ${\Delta \%P}_t-{\Delta \%F}_t\approx 0$. This goal is a well-established practice in portfolio management, where the objective of a fund, institution, or individual is modeled against a benchmark. Typically, this benchmark is a stock, futures, or fixed income index, such as the Dow Jones index or the Standard & Poor’s (S&P) Goldman Sachs Commodity Index (SPGSCI). Following the Global Investment Performance Standards (GIPS) and the well-established literature on investment management, such as Maginn et al. [22], there are two primary types of portfolio management: passive and active. The former is a type of management where the investor or manager replicates the performance of a given index or benchmark (the goal in the quantitative method tested herein). The latter attempts to generate surplus returns or extra returns above those provided by the benchmark (a concept known as alpha). The purpose of this paper is to test the effectiveness of the portfolio selection method in reducing basis risk, which translates to the concept of tracking error in the terminology of portfolio management. Theoretically speaking, passive management implies zero tracking error (i.e., zero basis risk) and zero alpha.

To replicate the performance of a given benchmark, a portfolio selection method known as minimum tracking error portfolio selection is employed. This method was proposed by Grinold [19]. To appreciate the difference between conventional (no benchmark restricted) portfolio selection, the following expression is the optimal portfolio selection problem solved in the geometric locus of expected returns of the set of assets involved in the portfolio selection $E{\left(r_i\right)}^{\ast }\in [E\left(r_i\right) ,max E(r_i\left)\right]$ with $r_i=\Delta {\%P}_{i,t}$ for all the assets or securities in the portfolio or investment set [20]:

$$w^{\ast }w w^{\prime }\Sigma w$$

(5)

This is subject to the following:

• $w^{\prime }e=E{\left(r_i\right)}^{\ast }$
• $w^{\prime }1=1$
• $w\ge 0$

In the previous expression, $e=\left[E\right(r_i\left)\right]$ is the asset-specific expected returns vector, and $\Sigma$ is the variance–covariance matrix, a key parameter to incorporate the benefits of diversification to portfolio variance (${\sigma }_p^2=w^{\prime }\Sigma w$) or risk exposure reduction.

To incorporate the restriction of the benchmark weights, $w_b$, to get the active weights, $w=w^{\ast }-w_b$, the optimal selection problem in (5) changes as follows:

$$w^{\ast }w w^{\prime }\Sigma w-2w^{\prime }\Sigma w_b$$

(6)

This is subject to the following:

• $w^{\prime }e=E{\left(r_i\right)}^{\ast }$
• $w^{\prime }1=1$
• $w\ge 0$

Even if it is feasible to estimate (6) using quadratic programming methods with current computer capabilities, the main drawback for its intended use in this paper is the requirement of prior knowledge of the benchmark portfolio weights, $w_b$. The goal of the portfolio selection method in this paper is not to create alpha from a theoretical benchmark that replicates the Mexican white corn price. The idea is to create a proper portfolio that replicates such a price. Consequently, the selection problem (6) is not helpful for the intended purposes herein. Setting aside Roll’s [21] critique of the mean variance inefficiency of the minimum tracking error portfolio selection, the inconsistency of this method through time, and the impact that the selected benchmark has in the mean variance results [22], the test of this paper does not use (6) because it is necessary to have a previous definition of the benchmark’s positions ($w_b$). Additionally, the only input to replicate is that of a single time series: the Mexican white corn price. Consequently, the authors used a method of asset–liability management practices known as the surplus efficient frontier [23,24]. This method was initially proposed by Sharpe and Tint [19] for the case in which the benchmark weights are unknown and the portfolio manager only has a percentage variation in the benchmark. For this purpose, the authors suggested estimating the risk premia or surplus of each asset of the portfolio, related to the benchmark ($surplu{s}_i=r_{i,t}-r_{b,t}$). With these surpluses, the manager can perform an estimation of the parameters $e_{surplus}$ and ${\Sigma }_{surplus}$ and estimate the efficient frontier (the geometric locus of the set of portfolios that satisfy (5) for ${\left(r_i\right)}^{\ast }\in [E\left(r_i\right) ,max E(r_i\left)\right]$) by solving the optimization problem in (5). The main difference between the minimum tracking error method in (6) and the surplus efficient frontier method is that it is not necessary to know the benchmark weights, $w_b$, only the benchmark’s return time series. If an asset manager only wants to replicate the performance of a benchmark, they only need to estimate the minimum variance portfolio along the surplus efficient frontier, that is, the portfolio on the surplus efficient frontier with the lowest variance. This leads to solving the following optimization problem, which is the particular case of (5) without the given portfolio’s expected return ($E{\left(r_i\right)}^{\ast }$):

$$w^{\ast }w w^{\prime }\Sigma w$$

(7)

This is subject to the following:

• $w^{\prime }1=1$
• $w\ge 0$

From another perspective, if the portfolio manager aims to create alpha through active management, she only needs to select one of the portfolios located in the upper part of the surplus efficient frontier (above the minimum variance portfolio that is the benchmark in this analysis). Figure 5 illustrates the portfolio selection in a mean variance context using simulated data for two normally distributed assets and a theoretical benchmark, which consists of 20% of the first asset and 80% of the second.

The figure plots the mean variance efficient frontier with short sales restricted, estimated with the optimization problem (5). Also, the figure depicts the efficient frontier of the portfolios estimated using the surplus efficient frontier estimated with $e_{surplus}$ and ${\Sigma }_{surplus}$. Because the simulated benchmark is a linear combination of the simulated assets, it lies on the efficient frontier, specifically at the tangency portfolio or the one with the best Sharpe ratio. In the real estimation of the surplus efficient frontier, the benchmark is not mean variance efficient and lies below the efficient frontier. Despite this, it is of interest, in both efficiency scenarios, that the benchmark is the portfolio with the lowest tracking error. The portfolios on the right of the benchmark have a higher tracking error. The surplus efficient frontier portfolio optimization is primarily used when a portfolio manager seeks to replicate or outperform a given benchmark or in cases involving asset–liability management.

For the specific case of basis risk reduction, the surplus efficient frontier portfolio selection reduces the process to selecting the portfolio with the lowest tracking error, a portfolio that, after estimating $e_{surplus}$ and ${\Sigma }_{surplus}$, solves the minimum variance portfolio selection problem in (7). Consequently, by selecting a set of futures, the corn price hedger can choose a portfolio with the lowest tracking error, using the Mexican white corn return (percentage price variation $\Delta \%P_{corn,t}$) as the benchmark to be replicated.

As noted in the previous description of this replicating futures portfolio, the core idea is to determine the optimal hedge ratio ($w_i$) in each future to reduce the tracking error or basis between the futures portfolio and the price used as a benchmark. That is the main rationale that motivates the test presented in this paper: using a portfolio of the most traded agricultural futures will allow us to replicate the spot price of Mexican white corn, which can be used for hedging purposes.

The minimum variance portfolio selection method, or its univariate benchmark-related version (surplus efficient frontier), has been tested primarily in equity markets or applications of asset–liability management; however, little has been written on its intended cross-hedging purposes. Only the work of Goswami et al. [25] takes a similar approach, testing the hedging effectiveness of corn, soybeans, and wheat by incorporating the impact of convergence and non-convergence between the spot and futures prices. The authors found a prolonged period of non-convergence between 2005 and 2011 and demonstrated that the lack of carryover results in low hedging effectiveness is due to increased price volatility.

As a key reference, by performing a quantitative cross-hedging of the Mexican Hass avocado price with a minimum tracking error futures portfolio, De la Torre-Torres et al. [15] backtested the minimum tracking error portfolio of 127 combinations (portfolios) of cocoa, coffee, yellow corn, wheat rough rice, soybean, and sugar futures. With weekly data from January 1998 to September 2023, the authors found that a coffee–sugar futures portfolio shows 0.94 hedging effectiveness.

Other studies have examined the effectiveness of hedging a future spot position using alternative methods for estimating hedge ratios. A widely used method is the long-term cointegration method proposed in Alexander [26] and Alexander and Dimitriu [27]. This method involves estimating the long-term cointegration relationship (coefficients or ${\beta }_i$ parameters) between the spot and the future position or even a portfolio of such futures. These coefficients must be estimated so that their sum equals 1 (the budget restriction in an optimal portfolio selection problem). The authors showed that this method is quite helpful and simpler than the optimal selection problem in (7).

For the case of agricultural cross-hedging, the work of Ziegelbäck and Kastner [28] shows that the cointegration method of Alexander and Dimitriu helps hedge the price of European live pigs with US lean hogs’ futures. Additionally, da Costa and Doner [29] employed this method, utilizing Markov-switching cointegration tests, to investigate the diversification effect between agricultural futures and equities. Their results indicate that, in a high-volatility context, the diversification effect diminishes, and the cointegration between the two markets is pronounced.

Despite the mathematical simplicity of the portfolio cointegration selection method, a necessary condition must be satisfied: the time series of the spot and futures prices must be cointegrated. This implies that they must be non-stationary, a situation that does not happen in some agricultural futures tested in this paper. Consequently, the surplus efficient frontier selection method was used in this paper. Other authors have tested the benefits of cross-hedging practices in agricultural markets using either the two previous methods or similar approaches.

Kumar and Pandey [30] tested the hedging effectiveness of Indian soybeans, corn, castor seeds, and guar seeds against futures in gold, silver, aluminum, copper, zinc, crude oil, and natural gas, all traded in India. The authors employed vector error correction models (VECMs) and constant correlation generalized autoregressive heteroskedastic (CCGARCH) models to estimate the hedge ratios. Their results show that agricultural futures exhibited hedging effectiveness between 0.3 and 0.7, while non-agricultural futures showed a value of 0.2. After using similar CME and NYMEX futures, their hedging effectiveness improved. Their conclusions suggest that the lack of hedging effectiveness is due to liquidity and contract specification heterogeneity, resulting in lower hedging effectiveness compared to the backtests of the CME and NYMEX futures. Similarly, Gupta, Choudary, and Agarwal [31] employed VEC and vector autoregressive (VAR-GARCH) models to examine the hedging of Indian agricultural commodity prices using energy and metals futures. The authors found that these futures led to poor hedging effectiveness if the settlement period was short. The opposite happens when the futures expire later. This result is consistent with the work of Rout, Das, and Rao [32], who also found low hedging effectiveness between Indian agricultural products and their corresponding futures.

In the case of Chilean cattle prices, the work of Troncoso-Sepúlveda and Caba-Monje [33] tested the use of CME cattle futures. Using the Johnson [34] and Stein [35] hedging method (a future spot portfolio selection problem like the one tested herein), the authors found that the hedging effectiveness was appropriate in this case. They also suggest extending the test to other staples, such as grains, milk, or pork.

In the Colombian case, Barrera, Cañon, and Sanchez [36] tested the hedging effectiveness of several agricultural prices using Colombian electricity futures. In only 9 of the 93 agricultural products tested, the authors found a significant hedging effectiveness of no more than 0.32. For this purpose, the authors employed VAR, VEC, and GARCH models to estimate the short- and long-term relationships, with the GARCH models yielding the best fit, a result expected due to the lack of cointegration between agricultural prices and electrical futures.

By incorporating prudence and temperance in the hedging decision process, Kamdem and Moumoni [37] used lower moments and ordinary least squares (OLS) and GARCH models in a time series to estimate the hedge ratio. Their dynamic model led to better hedging effectiveness in storable commodities (the carryover effect).

Related to the use of the EU IST tool and the use of futures to hedge Italian agricultural products, Penone et al. [38] tested the hedging effectiveness of soybeans, corn, and wheat using Euronext or CME futures. By employing a naïve hedging strategy (a 1-to-1 spot minus future position), the authors found that the OLS or GARCH hedge ratio method yields better hedging results with Euronext futures. The authors concluded that, using the previous hedging method, the European futures market is more suitable for hedging purposes in Italy.

For the specific case of the Mexican price market, Barrios-Puente et al. [39] applied the binomial tree theory to estimate the spot price over several hedging periods (ranging from one to four months) and utilized the NYMEX coffee futures. Their results suggest that hedging the Mexican price of coffee with futures and their statistical method leads to higher income, especially if the hedging horizon is longer.

Finally, the work of Erasmus and Geyser [40] tested the use of the CME soybean futures contract, denominated in South African rand and traded on the Johannesburg Stock Exchange (JSE), to hedge the soybean spot price in the same exchange. By testing the OLS, VEC, and VEC-GARCH models (after conducting the necessary cointegration tests), the authors found that the future contract outperforms the spot price when the prices are close to or below the export soybean price.

As noted in the literature review, cross-hedging practices lead to better income results for commodity and non-commodity sellers (both producers and intermediaries). Additionally, it has been noted that the CME yellow corn future and the Mexican white corn spot prices exhibit no significant relationship, and the former serves as a poor hedge for the latter when used as the sole hedging security. Finally, previous works have shown the benefits of using a minimum tracking error futures portfolio (selected within a surplus efficient frontier context) to replicate or cross-hedge spot positions.

Consequently, these previous works and results motivate this paper because the authors extended some of these to prove that using not a single CME corn future position, but a mimicking portfolio, leads to better hedging effectiveness (corn price replication).

The previous literature review highlights the selection of the surplus efficient frontier as the optimal portfolio selection tool. As noted in these earlier works, these tested the benefits and hedging effectiveness of either a single future or a multiple portfolio selection in cross-hedging. From the main models to achieve this goal, there are four that could be a potential solution:

• Minimum variance liability portfolio selection as in (6).
• The surplus efficient frontier with previously estimated surpluses or deficits of the basis between the ith futures and the benchmark or spot price of interest. This uses a model as in (7).
• The cointegrating VEC model suggested by Alexander and Korovilas [41] where the long-term equation’s normalized coefficient gives the investment level in each portfolio.
• The Elam et al. [17] multivariate regression method with the residuals of the future benchmark (spot price) basis as in (4).

From these, the authors selected the second method (surplus efficient portfolio selection) for four main reasons:

• Using the minimum variance liability portfolio selection requires a benchmark with a market or reference investment level in each future. The purpose of this paper is to determine which futures portfolio and which investment levels are necessary for such a benchmark. That is, the purpose of this paper is to design an agricultural futures benchmark to replicate the Mexican white corn spot price.
• The VEC model portfolio selection assumes that the spot price and the futures are not stationary and, consequently, cointegrated. As noted in the results section, some futures show no unit root, leading to the conclusion that this method is not feasible for the staple of interest.
• Also, the estimation of the VEC model, as Alexander and Korovilas [41] suggest, requires the solution of a quadratic problem in which $\sum {\beta }_i=1$. In a scenario in which there is no cointegration, the short-term estimation of such regression models could be estimated with a short-term VAR model that could lead to a similar solution as in the surplus efficient frontier, with a less complex (multiple equation) estimation method where $\sum w_i=\sum {\beta }_i=1$. Consequently, the surplus efficient frontier method and the VAR model could lead to similar solutions. This is an assumption to test in other spaces and is out of the scope of this paper.
• The Elam et al. [17] model is an intuitive method and an extension of the single-equation model for portfolio selection [42,43,44]. Despite this, it is necessary to test and compare if the investment levels with this method lead to better price replication (less basis) than the surplus efficient frontier.
• Finally, the surplus efficient frontier (and the minimum variance portfolio) is a method of portfolio selection established by Markowitz. This comprises a theoretical corpus of methods that is still used in the industry [45].

Based on this literature review and the related motivations, the following section outlines the data gathering and processing method, as well as the main results and findings of the backtests.

3. Methodology

3.1. Data Gathering and Processing

To test the working hypothesis, the authors obtained the weekly white corn price for the traded kilograms in the main public markets across the 32 states of Mexico [46]. With these 32 trade prices, the authors estimated a weekly mean value to obtain the mean price value of white corn in Mexico ($P_t$). This price will be used herein as the spot price to be hedged. Figure 6 shows the box plot of the historical prices of the 32 states (origins) of white corn. It is noted that the price difference is low among states, with the exception of Aguascalientes and Quintana Roo (who are not leading producers of corn, given their ecological conditions). This result, and following the findings of Ortiz-Arango and Montiel-Guzman [6], who also found similar price behavior, suggests that the assumption of a homogeneous and single (mean) white corn price in Mexico can be upheld. The test of this assumption and the impact of market frictions is left for further research.

With the historical corn prices, $P_t$, the authors estimated the continuous-time returns (log differences or $r_{P_t}$) for both the surplus efficient frontier portfolio selection and the hedging effectiveness test.

Similarly, the authors retrieved the weekly historical closing prices of the ten most traded agricultural futures from Refinitiv databases [48] for CME and NYMEX. To account for the impact of the Mexican peso–US dollar (USDMXN) FX rate, the authors converted the original price in the futures units’ specifications to the Mexican peso per kilogram equivalent. For example, the yellow corn contract in the CME specifications is quoted in US dollars per bushel, and the contract units are 5000 bushels. For this purpose, the authors divided the price by 100 (the quote is in US cents per dollar) to convert the price to US dollars and then divided this price by 25.4 (a CME corn bushel, equivalent to 25.4 Kg). With this US dollar per kilogram price equivalent, the authors multiplied this price by the current USDMXN rate at t to arrive at the Mexican peso per kilogram equivalent price of the future contract.

Table 1. The futures used in the backtests, their general contract specifics, and the steps followed to express the traded price in US dollars per kilogram.

Future	Ticker in This Paper	Refinitiv RIC	Contract Unit	US Dollar Quote Conversion in the Paper	Kilogram Transformation in This Paper	Future Contract Standard Units	Future’s Exchange
1-month corn	cornFuture	Cc1	Bushel	Price/100	Price/25.4	5000	CME
1-month wheat	wheatFuture	Wc1	Bushel	Price/100	Price/27.21	5000	CME
1-month rough rice	roughRiceFuture	RRc1	Hundred weight (cental)	None	Price/45.36	2000	CME
1-month soybean	soyBeanFuture	Sc1	Bushel	Price/100	Price/27.21	5000	CME
1-month oats	oatsFuture	Oc1	Bushel	Price/100	Price/27.21	5000	CME
1-month cocoa	cocoaFuture	CCc1	Metric ton	None	Price/1000	10	NYMEX
1-month coffee	coffeeFuture	KCc1	Pounds	None	Price × 0.453592	37,500	NYMEX
1-month no.11 sugar	sugar11Future	SBc1	Pounds	None	Price × 0.453592	112,000	NYMEX
1-month cotton	cottonfuture	OJc1	Pounds	None	Price × 0.453592	50,000	NYMEX

Source: Own elaboration with data from CME [46].

summarizes the futures contracts used in the backtests, the ticker used to identify them in this paper, the Refinitiv identifier code (RIC), the contract trade unit, the conversion operation used to convert the price to US dollars, the conversion operation used to express such prices in US dollars per kilogram, the standard number of units traded per contract (contract size) and the exchange where the future trades.

With these future price data, $F_{i,t}$, the authors also estimated the continuous-time return, $r_{F_{i,t}}$. As the first test, the authors examined the hedging effectiveness, OLS hedge ratio, and Engle–Granger [7] cointegration tests of each future’s return $r_{F_{i,t}}$ with the Mexican spot rice price. The core idea is to prove that not all futures are cointegrated with the corn price and, following Ortiz-Arango and Montiel-Guzmán, to demonstrate that the hedging effectiveness of a single future position is not appropriate.

The historical data for both white corn and futures are weekly data from January 1998 to February 2025.

Using the historical returns data, the authors estimated the optimal futures’ investment weights, $w^{\ast }$, of the surplus efficient frontier in (7). Given $w^{\ast }$ at t, the authors estimated the percentage price variation in the simulated portfolio as follows:

$$r_{port,t}={w^{\ast }}^{\prime }r, r=\left[r_{i,t}\right]$$

(8)

The rationale of the backtests and the working hypothesis is that the ideal futures portfolio must have a percentage price variation equal to that of the Mexican white corn:

$$H_0:r_{port,t}=r_{P,t}$$

(9)

To test this equality, the authors estimated the optimal portfolio from 1 January 2000 to 23 February 2025 and used the weekly historical returns data from 2 January 1998 to the date of the simulation.

To determine which agricultural futures portfolio is the best performing in terms of price replication, the authors simulated 1013 combinations (portfolios) of the ten futures. The best hedging portfolio will be the one with the highest hedging effectiveness, estimated as in (1). To strengthen these results, the authors performed a cointegration test of the simulated portfolio with the avocado price.

Finally, to demonstrate the practical application of these replicating portfolios, the authors backtested the top five best-performing portfolios. The backtest involves taking a short position in the simulated portfolio if a down price trend is expected at $t+1$. To identify such a downtrend, the authors employed a technical analysis indicator widely accepted among investors: the moving average convergence–divergence (MACD) [49]. This indicator (a lag indicator) suggests that if its value is positive (negative), an upward (downward) price trend is expected. The MACD is calculated as the difference between the 26-day moving average of the Mexican white corn price and the 12-day moving average.

The authors’ selection of the quantitative trading rule is arbitrary and was chosen because this indicator is easy to estimate and is widely used in the financial industry. The use of other technical indicators, machine learning, or quantitative methods for the hedging decision rule is left for further research. The authors used this technical indicator to conduct an initial test of the practical benefits of the corn price replication method proposed herein.

To determine if the simulated agent must hedge its corn price position, the authors used three scenarios:

• To hedge (buy a short portfolio position) if the MACD is lower than zero.
• To hedge if the MACD is lower than its signal line (9-day mean value of the MACD).
• To hedge if the MACD is lower than zero and its signal line. For the full backtest, the authors used the following assumptions:
  • There are no liquidity issues in the traded futures.
  • The hedger will use only the ten agricultural interests herein.
  • There are no trading costs.
• The Mexican peso stock value in kilograms to hedge is equal to the notional value of the futures contract in the portfolio, excluding the impact of the hedged value on basis risk.
• The covariance matrix used in (7) is the conventional time-fixed covariance, leaving the use of dynamic, non-linear, and time-varying covariances for further research.
• The foreign exchange (FX) impact is low because the simulated agent has preferential access to low transaction costs.

The first reason for using only the ten most traded US agricultural futures is the dimensional feasibility of the quadratic programming problem in (7) when the number of futures increases. Due to this limitation, some portfolios or combinations were excluded from the simulations due to feasibility issues. The second reason for using such a limited futures set is that these backtests serve as a first approach to test the benefits of using the portfolio replication of interest. The authors kept the agricultural nature of the futures, setting aside the use of non-agricultural and even non-US futures for further research.

As an underlying assumption and following the backtest for a U.S. stocks and VIX futures strategy in [50,51], the authors did not include the impact of trading costs in the simulated portfolios or the trading rule for three key reasons:

• De la Torre-Torres et al. [50,51] showed that the impact of trading fees of an actively managed portfolio of stocks or volatility futures is negligible in weekly trading decisions, a result that the authors assumed to be true.
• The impact of slippage, liquidity (bid–ask spread), or other trading costs is also negligible because the price-mimicking portfolio is rebalanced on a weekly periodicity.
• The number of trades in the simulated weekly portfolios is small, as noted in the results section, because the investment levels do not change significantly from week to week.

Departing from these reasons, the impact of trading costs in the simulations will be left for further research due to the complexity of the analysis and because this paper’s primary goal is to prove the appropriateness of the quantitative method for white corn price replication.

3.2. Discussion of Results

3.2.1. Individual Future’s Cointegration and Hedging Effectiveness Tests

Before examining the results of the 1013 combinations of futures or portfolios, it is essential to review the hedging effectiveness and cointegration tests of the individual future-specific tests with the Mexican white corn price. Figure 7 shows the historical Mexican peso per kilogram price of both the spot and CME futures prices. The upper panel displays the price of the Mexican peso per kilogram, and the lower one shows the continuous-time return. At first glance, the historical data suggest that the prices may be cointegrated based on the close historical performance of both time series. To test this potential long-term relationship,

Table 2. Unit root tests’ p-values for the price (level) time series of the Mexican white corn and the futures of interest.

Time Series	Augmented Dickey–Fuller Test	Phillips–Perron Test	KPSS Test
Mexican white corn price	0.5835	0.7995	0.0100
cornFuture	0.0981	0.0747	0.0100
wheatFuture	0.0868	0.0100	0.0100
roughRiceFuture	0.0100	0.0100	0.0100
soyBeanFuture	0.0311	0.0539	0.0100
oatsFuture	0.0422	0.0242	0.0100
cocoaFuture	0.9900	0.9900	0.0100
coffeeFuture	0.9900	0.9900	0.0100
sugar11Future	0.0452	0.0124	0.0100
cottonFuture	0.3246	0.5565	0.0100
orangeJuiceFuture	0.3246	0.5565	0.0100

Source: Own elaboration with data from Refinitiv [47].

summarizes three cointegration tests: the Augmented Dickey–Fuller [52] with one lag in the residuals of the auxiliary test’s regression, the Phillips–Perron test [53], and the Kwiatkowski et al. [54] (KPSS) test.

The first two tests’ p-values correspond to a unit root test null hypothesis and the third to a stationary trend vs. non-stationary time series. As noted in Table 2, the Mexican white corn price has a unit root, and the corn and wheat futures have it at a 10% significance level (practically no unit root). These results align with those of Ortiz-Arango and Montiel-Guzman [6], who found weak cointegration and a non-significant relationship between the CME corn futures and the origin-specific white corn prices. The other futures of interest have no unit root except for rough rice, soybeans, oats, and sugar, which have it at a 5% significance level. Due to the lack of unit roots between some futures (starting with the one of primary interest in this paper: corn), it is not possible to determine if the Mexican white corn and these futures are cointegrated. Consequently, using other hedging methods such as the one suggested by Alexander and Dimitriu [27] is not feasible. Consequently, the use of a surplus efficient frontier portfolio selection (7) with two or more futures in the replicating portfolio is a potential solution.

To check for robustness and the potential impact of structural breaks, the authors performed recursive unit root tests with an increasing time series from 2 January 1998 to 1 January 2000, as the first test date. Then, the authors extended the time series length to the following date, up to 23 February 2025. Figure 8 shows the historical p-values (multiplied by 100) of the three-unit root tests of Table 2. The first panel displays the p-values of the Augmented Dickey–Fuller test, while the two subsequent panels show the p-values of the Phillips–Perron and KPSS tests, respectively.

As noted in this figure, the p-values are not stable through time. In some periods (like the ones after the 2007 financial crisis or the 2020–2021 period related to the COVID-19 turmoil in financial and commodity markets), the p-values of all the time series (except for rough rice) showed unit roots. Out of these periods, the time series shows a p-value near or even higher than 10% (in almost all dates). This figure supports the previous conclusion derived from Table 2. At this point of the review of the results, it could be of interest to perform a unit root test to incorporate the presence of structural breaks or regimes. Despite this need, the use of the proper test and its rationale is left for further research due the complexity of such a task.

3.2.2. Mexican White Corn Price Replication with the Futures Portfolios

To review the result of the simulations of the 1013 portfolios, it is essential to mention that the name of the futures as it identifies each simulated portfolio. For example, the portfolio “cornFuture, oatsFuture” is a simulated portfolio that includes only the corn and oat futures. Also important to mention is that 256 portfolios (25.2771% of the 1013 simulated) were not feasible to test due to dimension issues in the optimal selection problem in (7). Of the 757 remaining, they were backtested. From these, the authors selected the five portfolios with the highest mean hedging effectiveness estimated as in (1). As a methodological note, it is important to mention that the five portfolio selection is arbitrary, and using a wider set, such as the ten best performing, leads to similar conclusions.

Table 3. Hedging effectiveness and hedge ratio summary of the five best hedging portfolios.

Simulated Portfolio (Futures Included)	Mean Hedging Effectiveness	Last Observed Hedge Ratio	Last Observed Hedge Ratios’ p-Value	Mean Observed Hegde Ratio	Mean Observed Hegde Ratio’s p-Value
cornFuture, oatsFuture	0.6202	1.0580	0.0000	1.2476	0.0000
cornFuture, wheatFuture	0.6180	1.0771	0.0000	1.3083	0.0000
cornFuture, orangeJuiceFuture	0.5687	0.8927	0.0000	1.1292	0.0000
cornFuture, sugar11Future	0.5639	0.9431	0.0000	1.0734	0.0000
cornFuture, cottonfuture	0.5559	0.9466	0.0000	1.1237	0.0000

Source: Own elaboration with data from Refinitiv [47].

summarizes the results. As noted in the exhibit, the portfolios with either corn and oat or corn and wheat futures are those with the highest mean observed hedging effectiveness. Even if this hedging effectiveness is good to replicate, there are some issues to address to achieve more accurate price replication. Consequently, the use of other types of futures, such as energy, metals, finance, or even weather, could be a potential solution to test in future works.

Although the two best-performing portfolios demonstrated a hedging effectiveness of 0.62 at most (leaving 0.38 as potential basis risk), it is necessary to determine whether these five future portfolios have a long-term relationship and whether their use remains appropriate for hedging purposes.

Table 4. Unit root and cointegration tests of the five best hedging portfolios.

Simulated Portfolio (Futures Included)	Augmented Dickey–Fuller Test	Phillips–Perron Test	KPSS Test	Phillips–Oularis Test (Naïve Strategy)
cornFuture, oatsFuture	0.5677	0.6331	0.0100	0.0100
cornFuture, wheatFuture	0.5622	0.4074	0.0100	0.0100
cornFuture, orangeJuiceFuture	0.5919	0.3650	0.0100	0.0100
cornFuture, sugar11Future	0.5842	0.4972	0.0100	0.0100
cornFuture, cottonfuture	0.5835	0.5232	0.0100	0.0100

Source: Own elaboration with data from Refinitiv [47].

shows the results of the three-unit root tests of Table 3, applied to the five best hedging portfolios. The last column shows the p-value of the Phillips–Ouliaris [56] cointegration test, an extension of Dickey–Fuller’s with a non-parametric Phillips–Perron unit root test in the residuals of the cointegrating equation.

As shown in Table 4, the prices of these five simulated portfolios are non-stationary and cointegrated. This result extends the previous tests by demonstrating that including more futures in the corn future position reduces noise and leads to a cointegrated time series with the Mexican white corn price. This first result suggests that it is feasible to replicate the corn price and use such a portfolio for hedging purposes. The only drawback to be tested is that the mean hedging effectiveness of this replicating (hedging) portfolio is around 0.62.

To test if these five best hedging portfolios are a proper option that can be used to hedge the Mexican white corn price, the authors performed the backtest detailed in the previous section by using four hedging rules or strategies:

• A naïve hedging method in which, each week, the theoretical agent (seller) of corn buys a short position in the replicating portfolio to hedge against price fluctuations at $t+1$.
• An $MACD$ strategy in which the agent hedges at t + 1 only if the $MACD<0$.
• An $MACD$ strategy in which the agent buys the short position in the portfolio if the $MACD$ indicator shows a lower value than its signal line (the 9-day moving average value of the $MACD$).
• A hedging strategy that combines the two previous ones.

To test the benefit of such a strategy, the backtest recorded the weekly profit or loss (P/L) of the simulated agent of the portfolio’s position (from $t-1$ to $t$). E.g., if the white corn price falls (grows) from $t-1$ to $t$, and the hedging portfolio does the same, the short position will pay a positive (negative) P/L that is added to the income of the price of each Kg of white corn sold in the spot market. To illustrate the results of the extra revenue generated by the hedging strategy (P/L), Figure 9, Figure 10, Figure 11 and Figure 12 display the historical accumulated income corresponding to the observed extra P/L values.

As noted, the naïve hedging strategy results in a volatile and inconsistent accumulated income through the backtest. Only the portfolios with corn and cotton, corn and oats, and corn and wheat futures generate additional income at the end of the test.

From the remaining three strategies, only the third strategy (when the $MACD$ is higher than its signal line) generates value in the corn–oats and corn–wheat portfolios, with the latter yielding the highest revenue.

These results suggest that using these two portfolios in conjunction with the third hedging strategy creates value and hedges a white corn seller (producer or intermediary) against negative market changes.

To have a general idea not only of which futures to use in the best replication portfolios but what the specific weight in each is,

Table 5. Mean investment level in five best hedging portfolios (values in %).

Future/Portfolio	cornFuture, cottonfuture	cornFuture, oatsFuture	cornFuture, orangeJuiceFuture	cornFuture, sugar11Future	cornFuture, wheatFuture
cocoaFuture	0.0000	0.0000	0.0000	0.0000	0.0000
coffeeFuture	0.0000	0.0000	0.0000	0.0000	0.0000
cornFuture	57.7016	64.2135	57.7016	57.8440	51.6741
cottonfuture	42.2984	0.0000	0.0000	0.0000	0.0000
oatsFuture	0.0000	35.7865	0.0000	0.0000	0.0000
orangeJuiceFuture	0.0000	0.0000	42.2984	0.0000	0.0000
roughRiceFuture	0.0000	0.0000	0.0000	0.0000	0.0000
soyBeanFuture	0.0000	0.0000	0.0000	0.0000	0.0000
sugar11Future	0.0000	0.0000	0.0000	42.1560	0.0000
wheatFuture	0.0000	0.0000	0.0000	0.0000	48.3259

Source: Own elaboration with data from Refinitiv [47].

summarizes the observed mean investment levels in each future of the five best hedging portfolios.

To strengthen the selection of the best hedging portfolio, the authors compared, in the polar plot of Figure 13, fifteen parameters of the five best hedging portfolios:

• The mean value of the hedging effectiveness (from Table 3), calculated for each date of the backtest (meanHedgingEffectiveness).
• The mean value of the hedge ratio of the simulated portfolio with the corn price (meanHedgeRatio).
• The mean Squared Error, that is the numerator of the hedging effectiveness as in (1) (meanRMSE).
• The accumulated extra income of the naïve hedging strategy (cumPLStrategy1).
• The accumulated extra income of the $MACD<0$ hedging strategy (cumPLStrategy2).
• The accumulated extra income of the $MACD<signal$ hedging strategy (cumPLStrategy3).
• The accumulated extra income of the $MACD<0,$ and $MACD<signal$ hedging strategy (cumPLStrategy4).
• The mean value of extra income at t of the naïve hedging strategy (meanPLStrategy1).
• The mean value of extra income at t of the $MACD<0$ hedging strategy (meanPLStrategy2).
• The mean value of extra income at t of the $MACD<signal$ hedging strategy (meanPLStrategy3).
• The standard deviation of extra income at t of the $MACD<0$, and $MACD<signal$ hedging strategy (meanPLStrategy4).
• The standard deviation of extra income at t of the naïve hedging strategy (meanPLStrategy1).
• The standard deviation of extra income at t of the $MACD<0$ hedging strategy (meanPLStrategy2).
• The standard deviation of extra income at t of the $MACD<signal$ hedging strategy (meanPLStrategy3).
• The standard deviation of extra income at t of the $MACD<0$, and the $MACD<signal$ hedging strategy (meanPLStrategy4).

As noted in Figure 12, practically all the best hedging portfolios show similar performance. The two key indicators of these are the accumulated extra income (a more sample-prone indicator) and the mean extra income at t. This last indicator shows the central value of the benefits generated by the short (hedging) portfolio position. From Figure 13 the reader can conclude that there is no difference in terms of performance between these and the selection criteria reduced to the accumulated extra income of the simulated portfolio and the performance shown in Figure 10 that suggest the using a portfolio of corn and wheat futures (with a mean investment level of 51.6741% and 48.3259%, respectively) is the best futures portfolio to replicate and hedge the Mexican white corn price and to generate an accumulated return of MXN 5.7664 per each kilo sold during the backtest. This result implies that a farmer who traded corn would have MXN 5766.4000 per traded ton each week during the simulation.

As noted from these results, it is feasible to replicate the Mexican white corn price to offer a minimum price scheme to corn producers. Consequently, this paper shows the first results for the Mexican white corn price in terms of price replication with an agricultural futures portfolio. The natural extension is to suggest and to test the implementation of a minimum price scheme. Unfortunately, such an endeavor is outside the scope of this paper and requires proper simulations and tests of the feasibility and financial profitability (and risk) for producers, traders and the government of a financial institution that could offer such scheme.

4. Concluding Remarks

Income risk hedging is crucial for enhancing food security in countries like Mexico. This is the case with staples that are key ingredients in the nation’s diet, such as white corn. Since the 1970s, Mexico has been in a stagnant production phase for this staple due to price reductions and more competitive prices abroad. This led to reduced benefits for small and medium-sized producers and a subsequent decline in supply. To minimize the impact of this result on inflation and food access at all income levels (mainly the lower ones), the Mexican Government has enrolled in several guarantee, strike, or minimum buy price programs of white corn financed with tax income. With such programs, the Mexican Government absorbed the shocks of buying at a price level and selling it to its citizens at a lower price, thereby seeking to stabilize the agricultural supply, consumption, and general prices. Because previous works have shown that hedging white corn prices with the yellow corn Chicago Mercantile Exchange (CME) futures is not feasible due to the lack of a long-term relationship between the spot and futures time series, this paper tests a potential solution with the price-mimicking portfolio.

Using a method widely employed in active portfolio management and asset–liability management, the authors tested the benefits of a portfolio of futures selected along the surplus efficient frontier (a special case of the minimum tracking error), a method to choose a portfolio with the lowest tracking error with the Mexican white corn price. The tracking error is defined as the variance (or standard deviation) of the difference in the return (percentage price variation) of the portfolio with that of the white corn.

After backtesting the performance of 1013 portfolios or combinations of the nine most traded agricultural futures of the CME and the New York Mercantile Exchange (NYMEX), the results suggest that using a portfolio invested in corn and wheat futures (with a 51.6741% and 48.3259% mean investment level, respectively) leads to a hedging effectiveness of 0.6180. Even if this value is not as close to 1 as it is preferred (a hedging effectiveness value close to 1 suggests an almost perfect income risk reduction), the improvements in terms of hedging are appropriate.

To test the practical use of this mimicking portfolio for hedging purposes, the authors backtested a trading decision in a naïve context, where a theoretical corn seller sold 1 kg of corn each week and hedged such sales with the mimicking portfolio in a t + 1 horizon. The other three simulated scenarios utilized the moving average convergence–divergence (MACD), a widely used and simple technical chart analysis indicator in the investing and trading practices. The results suggest that making hedging decisions when the MACD is below its signal line (its nine-day moving average) leads to a significant added value of MXN 5.7664 per kilo sold during the simulation (MXN 5766.400 per traded ton).

What this result implies for food security in practical terms is that the Mexican Government or a financial institution could offer a buy strike price $K$ at t + 1 to a given white corn seller (producer or intermediary) and balance or hedge the price risk with a short position of the simulated futures portfolio. With this hedging of the offered hedge, the Mexican Government would not need to use public resources to pay or absorb the loss incurred, due to the income generated with the short position in the mimicking portfolio. Consequently, this paper demonstrates the benefits of cross-hedging a non-commodity staple, such as Mexican white corn (or a similar product), with a portfolio of corn and wheat futures.

Among the results of this paper, it is essential to highlight the need of repeating the backtest in a context that incorporates trading costs, and most of all, the impact of the difference not in the portfolio and white corn price but between the nominal or face value (due the futures contract specifications) of such a portfolio with a Mexican peso equivalent face value. This test is of practical importance because, as is the case in the fixed income or foreign exchange markets, the Mexican Government or a financial institution that offers the hedge must pool the hedge prices of several corn sellers to balance them with the short futures position. Similarly, the use of other, more dynamic and time-varying covariance matrices in the portfolio selection problem in (1) is a necessary step.

In terms of diversification, utilizing more agricultural and non-agricultural futures to enhance hedging effectiveness is a natural and necessary step to verify the robustness of this paper’s results.

In terms of hedging decisions, testing other quantitative or artificial intelligence (AI) methods to forecast a decline in white corn price is a practical need to assess the usefulness of this price-mimicking portfolio. Also, to compare the use of the MACD for hedging decisions with the suggested agricultural futures portfolio against the use of other AI methods (or technical indicators) will be a necessary step for practical use.

Finally, related optimal portfolio selection models could be extended to incorporate the use of other quantitative or artificial intelligence methods to select the optimal or mimicking portfolio, particularly due to the dimension issues in the quadratic programming method used herein.