Non-Commodity Agricultural Price Hedging with Minimum Tracking Error Portfolios: The Case of Mexican Hass Avocado

Abstract

The present paper tests the use of an agricultural futures minimum tracking error portfolio to replicate the price of the Mexican Hass avocado (a non-commodity). The motivation is that this portfolio could be used to balance the basis risk that the avocado price hedge issuer could face. By performing a backtest of a theoretical avocado producer from January 2000 to September 2023, the results show that the avocado producer could hedge the avocado price by 94%, with the hedge offered by a theoretical financial or government institution. Also, this issuer could balance the risk of such a hedge by buying a coffee–sugar futures portfolio. The cointegrated or long-term relationship shows that using such a futures portfolio is useful for Mexican Hass avocado price hedging. This paper stands as one of the first in testing futures portfolios to offer a synthetic hedge of non-commodities through a commodities’ futures portfolio.

1. Introduction

Income risk reduction is among the main issues that farmers face because of climatological, market, or even social factors. This task is a concern that could even affect the subsistence of a farm or producer, especially if its size (and related economies of scale) is medium or small. This risk could even have an essential impact on the farmers’ country’s economy due to the financial sustainability of their production. Food security and even sovereignty are among the main elements a given country wants to enhance. A proper (stable) income for their producers is essential to this goal.

This paper will not discuss food security and sovereignty, because both terms and their implications still need to be debated. These concepts are mentioned because income risk affects these goals, endangering access to food for the population in each country and economic growth and development in the local economies.

Among the practices that could enhance farmers’ financial sustainability are insurance policies against weather or other catastrophe risk coverage, government (trade) subsidies, mutual funds to face risk, or price hedging with derivatives. The World Trade Organization (WTO) classifies some of these practices by color groups. Among the color groups of greatest interest are the green box practices (followed by the amber ones), which are considered among the most world-trade-friendly and have a low impact on price formation. Examples of these practices include the EU’s Income Stabilization Tool (IST) [1], which has been useful in hedging the farmer’s three-year mean income (or a five-year Olympic one) and has enhanced the production of several products in countries such as Austria, Croatia, Italy, Slovenia, and Spain, and also had other effects such as increasing income equality when the contributions to the mutual fund used to hedge potential income losses are proportional to the farmer’s income [2,3,4,5].

Other income risk hedging practices are considered “amber” by the WTO. Examples include the use of futures hedging or the payment of a minimum price to motivate agricultural production among small-scale farmers. Both practices and subsidies are among the most used tools in the U.S. and Canada. These practices and subsidies are among the main reasons why countries such as the U.S. are net exporters of agricultural products such as corn and wheat (among others).

With these practices, countries like Mexico need to promote food security, understood as the capacity to produce the necessary food with local producers and with no or low dependence on agricultural imports. Food sovereignty is an extension of food security, with a positive impact on the development of local production techniques [6,7,8,9,10].

The Mexican case is of interest because the Mexican government has made efforts to support small and mid-size producers’ income. Among the most recent public programs, the Mexican government created a government agency known as Mexican Food Security (SEGALMEX by its acronym in Spanish) that focuses on buying the small and mid-scale producers’ output in vital products such as white corn, beans, wheat, and milk. For the specific case of corn and beans, the Mexican government pays a floor or minimum price if the spot market price of these commodities is below the yearly yellow corn or wheat futures price in the Chicago Mercantile Exchange (CME). The payment of this floor price comes with resources from the Mexican government (taxpayers) paid through SEGALMEX, and the latter has an unbalanced (contrary) position to cover the price risk of the offered hedge. Despite its practical usefulness, SEGALMEX still needs improvements to include primary agricultural products [11,12,13,14,15] or staples, and it could also be helpful to extend its duties to other ones, such as avocados.

Among the main agricultural products in terms of exports, avocado has become one of the most notorious and fast-growing. Its exports to the U.S. increased thanks to a border opening in 2000 [16,17,18,19], and it has become one of the leading export producers in the country. More specifically, the premium Hass avocado species is exported the most, and its production has increased mainly in Michoacán. This state has ideal geological and weather conditions due to the location of avocado crops in places with volcanic soil at more than 2000.00 m from sea level. According to the APEAM (the Mexican Avocado Producers and Exporters Association), a certification office and lobby group, Michoacan is the leading avocado producer in Mexico and worldwide. Consequently, this economic activity is an essential vehicle for economic growth and development. Therefore, it is of interest to have proper price hedging to reduce income risk and to enhance economic growth and development in Mexico and Michoacán in particular.

Even if the Hass avocado is an essential staple in Mexico and a delicacy in places like the U.S., Europe, and Asia, it is not a commodity. Despite this, the fruit has experienced an exponential demand outside Mexico, contributing significantly to Michoacán’s gross domestic product (GDP) and Mexico’s. According to official figures of the Mexican government [20], the added value of avocado production has represented 3.58% of Michoacán’s GDP and 0.10% of Mexico’s. Figure 1 shows the historical relevance (regarding GDP) of such fruit in Michoacán and Mexico as a whole.

Departing from this fact, even if this fruit is not a commodity in the broad significance of its relevance in worldwide production and consumption (like corn or rice), its economic impact in this country (and Michoacán) is capital and needs a proper income hedging strategy to enhance producers’ income and GDP levels.

One of the main limitations of hedging the price of the avocado is the absence of a commodity derivative (such as a traded future or option). Therefore, using a synthetic hedge through a portfolio of the most traded agricultural futures is a potential answer tested herein. This paper tests the use of a minimum tracking error portfolio that seeks to replicate the performance of the Mexican spot avocado price. A tracking error portfolio is optimal for replicating the performance of a benchmark or statistical (economic) reference. Usually, this benchmark is a well-known security price index or multi-asset one. The main aim of this minimum tracking portfolio is to reduce the difference or distance between its percentage price variation and one of the benchmarks. In general terms, the tracking error of the return of a given portfolio ${TE}_{p,t}$ is the standard deviation of the difference between that portfolio’s return ($r_{p,t}$) and the benchmark’s ($r_{bm,t}$):

$${TE}_{p,t} = \mathsf{\sigma }\left(r_{p,t}-r_{bm,t}\right) = \mathsf{\sigma }\left(\sum w_i\cdot r_{i,t}-r_{bm,t}\right)$$

(1)

The tracking error is used to select a portfolio with optimal weights (${\mathbf{w}}_{\mathbf{i}}^{\mathbf{*}}$) that minimize ${TE}_{p,t}$. This is a practice common in asset–liability management (AL) or passive portfolio management investing [16,17,18,19,20]. The benchmark is either an economic factor or statistical index replicating the liability’s performance to hedge or a market portfolio (index). This paper uses the historical Mexican avocado price’s return (national average price at $t$) as the benchmark return ($r_{bm,t}$) to select the optimal (minimum tracking error $\mathbf{w}\mathbf{*}$) portfolio of agricultural futures to hedge (replicate) the avocado price.

The core idea is that if it is possible to select a minimum tracking error portfolio of agricultural futures that replicates the avocado price at $t$, a financial or government institution (such as SEGALMEX) could use it to hedge the Mexican producers’ avocado price and, as a consequence, to hedge the producers’ income without any market distortion, such as subsidies or the payment of an insurance’s prime.

Assuming that SEGALMEX or another public institution could buy this futures portfolio to hedge and sell price coverage to the producer, this paper aims to test whether the Mexican avocado producers’ income risk could be significantly reduced (hedged) because there is low or no basis risk.

If this result is feasible, Mexican financial institutions or the government could use this synthetic hedge (minimum tracking error futures portfolio) to sell avocado price hedges with the opposite (balanced) hedging position. SEGALMEX could buy the futures portfolio to transfer the cost of hedging the avocado price to the Mexican producer. The main question to address is whether the difference between the performance of the futures portfolio and the avocado producer (basis risk) is as low (near zero) as possible.

Following this motivation, this paper simulated the performance of 127 agricultural futures combinations to select a minimum tracking error portfolio. This portfolio performance was compared against the Hass premium quality avocado’s price. It also simulated the theoretical income a theoretical avocado producer would have had if they had bought the avocado price hedge in a hedging horizon of $t+1$, $t+4$, $t+12$ weeks. The main goal of this second motivation is to enhance the hedging effectiveness ($H{E}_{p,t}$) to values near 1, suggesting that this synthetic hedge or minimum tracking error portfolio is appropriate for income risk hedging purposes.

The authors’ theoretical position or working hypothesis is that using agricultural futures minimum tracking error optimal portfolios will create a synthetic hedge of the avocado price, leading to a significant income risk reduction for an avocado producer. This implies that an agricultural futures portfolio could replicate the Mexican Hass avocado price properly.

This paper’s related social and economic motivations are the benefits of reducing income risk to avocado producers, because this fruit’s production is the main agricultural product and creates value in 68 of the 113 municipalities in the state. This industry is one of the most significant for job creation and wealth generation and one of the leading activities in terms of economic development.

Among this paper’s financial theoretical and practical motivations, the leading motivation is to test the use of minimum tracking error agricultural futures to hedge the price of a non-commodity product, such as avocados.

Few studies in the literature test the use of commodity futures to hedge the price (and income) of non-commodities. This paper aims to fill this gap for the specific case of Mexican avocado production. A theoretical motivation is to use the minimum tracking error technique to hedge non-commodities.

Departing from this motivation, this paper’s results innovate in four ways. First, it is among the first papers to test the use of minimum tracking error optimal portfolio selection to manage an agricultural futures portfolio to hedge the price of an agricultural non-commodity. Second, it is among the first papers to show the benefits of using the price of a non-commodity (avocado) in an asset–liability optimal portfolio selection model [21,22,23,24,25]. Third, it gives quantitative suggestions and tests of the benefits of using market-traded futures to hedge the price of non-commodity agricultural products, such as avocados. Finally, the results could potentially be used for avocado price hedging (and similar non-commodities). These hedging tests could be helpful in policy recommendations and food security in countries or regions with similar income risk needs.

With this paper’s leading theoretical and practical motivations elaborated, the following section provides a brief literature review of the related works and results that support or motivate the present one. The third section briefly reviews the test method, depicting the minimum tracking error portfolio selection and mentioning the test algorithm and methods to test the working hypothesis. The fourth section discusses the main results, and the fifth one presents the concluding remarks and guidelines for further research.

2. Previous Literature Review

The previous literature testing the use of hedging spot commodity prices with futures is vast, but the most related to this paper deals with hedging agricultural spot prices with futures. Hedging strategies exist in countries where several exchanges or delivery spot markets exist. Cases such as India, Vietnam, and even the U.S. are interesting [26,27,28,29,30]. In these works, the main findings suggest that heterogeneity in the delivery spots (or markets) or the presence of several futures markets could lead to basis risk (difference in value between the spot hedged position and the hedging futures one).

Using income risk reduction or insurance could increase production due to more certainty in the agricultural product’s market price [7]. Several agricultural prices or income risk hedging programs exist in countries like the U.S. and in the EU. Authors like Roznik et al. [8] and Glauber [9,10] study the development and even theoretical properties of insurance or hedging decisions in the U.S. and Canada. Glauber [10] reviews the development of the U.S. methods for insurance risk hedging. U.S. farmers have three tools to hedge risk: (1) income insurance, (2) government subsidies, and (3) agricultural commodities derivatives (futures and options). Following Glauber [10], income insurance (paying a private insurance company’s risk premium) to hedge income due to natural catastrophes or even economic ones is one of the most used methods because these policies are subsidized in regions with high catastrophic risk. Government subsidies also apply when some products’ prices (like those of yellow corn) fall below a threshold. Finally, using futures or options to hedge a producer’s income is a liquid solution in some agricultural products such as cocoa, coffee, corn, random-length lumber, milk, oats, orange juice, rough rice, soybean, sugar, and wheat (among others). These futures and their corresponding options are mainly traded in securities exchanges, such as the Chicago Mercantile Exchange (CME), the Chicago Board of Options Exchange (CBOT), and the New York Mercantile (NYMEX). For the specific case of Canada and the U.S., using these derivatives does not distort international trade, but the particular case of subsidies to agricultural products does. Despite this, Glauber [9,10] and Roznik [8] show that these three price or income hedge methods have allowed the development of agricultural activity, thanks to a higher degree of income certainty in these markets. Also, these authors suggest that the cases of catastrophic risk, production, and income insurance subsidy premiums are a fiscal charge of billions of U.S. dollars.

Dos Santos et al. [31] produced a work review that discusses the most relevant papers on income hedging and hedging activities with futures. Their results suggest the relevance of income and enterprise risk hedging in distressed periods, a practical need tested herein for avocado producers’ income risk.

The EU’s Income Stabilization Tool (IST) [1,32] is another income risk mitigation mechanism. It is mainly subsidized by the European Union. It focuses on hedging the general income level below a farmer’s threshold estimated with the three-year mean income (or the Olympic five-year income average). This system is considered more appropriate for international trade standards (a green-label practice according to WTO) and subsidizes farmers’ yearly risk premiums paid to a mutual fund. If the income level each year falls below the threshold of 30% of that average income, the mutual fund’s proceedings are used to pay the difference. The European Union subsidizes the mutual fund deposits and the creation of mutual funds. The EU intends to cover several agricultural products.

The IST has been used mainly in Austria, Croatia, Italy, Ireland, Slovenia, and Spain. Authors like Severini [3,4] have studied the historical development of these systems in Italy. By performing a multivariate analysis and simulating the theoretical decision that motivates the use of the IST in Italy, Severini [3,4] found that differentiated mutual fund payments could be of interest to encourage the use of the IST or enhance farmers’ income.

The work of Čop et al. [2] studied the factors that motivate the use of the IST in Austria, Croatia, and Slovenia and found that the income threshold or the compensation level is among the main factors that motivate the use of this tool among wine producers. Similarly, Rippo and Cerroni [5] review the motivations for using the IST among apple producers and find that a more local mutual fund contribution scheme also leads to higher participation and a more significant interest in using the IST among Italian apple producers. Also, the authors show that belonging to a cooperative and understanding such schemes increases the use of the IST, and the Canadian or U.S. income hedging schemes are similar in their goals but different in their methods. The IST subsidizes the payment to a mutual fund, and the former makes direct payments to farmers and, in specific cases, subsidizes a private insurance company.

For the Mexican case, the one of interest herein, the efforts to support farmers’ income started in the 1930s. These efforts had little impact on creating economies of scale among small and mid-size farmers. Despite this and following a local industry protective trade policy (a fiscal policy more oriented to oil exports), Mexico saw an acceptable development of agricultural activity with an acceptable level of food sovereignty, that is, a farming activity with low dependence on food imports [12,13,33]. During the 1970s, Mexico experienced an abrupt shift due to fiscal deficits and high inflation levels. This led to income destruction among small and mid-size farmers, creating a food output crisis. To support food security and sovereignty, the Mexican government allowed more food imports and established a minimum price policy for agricultural products such as white corn, beans, and milk. With this minimum price policy, the Mexican government bought these products, stored them, and sold them to the Mexican population through government-sponsored stores at lower prices. The core idea of this policy was to incentivize small and mid-size agricultural production with more stable and appropriate minimum prices and to secure food distribution with lower prices among citizens [11]. This model was later evolved with other policies in which Mexican agricultural producers had market protection. Despite this taxpayers’ effort, food imports increased in the primary agricultural products, leading to higher food dependence on producers abroad and a concentration of the leading agricultural products in some states, the ones with the biggest economies of scale and the most appropriate natural conditions. As a result, the Mexican government evolved its food security program to one oriented to paying a minimum price for corn, beans, and milk. Nowadays, a public agency named Mexican Food Security (SEGALMEX) buys some agricultural products with a minimum price hedging (like white corn or beans). For the specific case of corn, the minimum price to hedge is estimated as the monthly average 1-month yellow corn future settle price in the Chicago Mercantile Exchange (CME), valued in Mexican pesos with the monthly average rate of the U.S. dollar–Mexican peso foreign exchange rate. The difference between the minimum price and the spot market one (if it is lower) comes from Mexican government fiscal proceedings. That is, tax contributors pay it.

No previous works deal with alternative public hedging methods for Mexican or non-commodity methods that reduce the burden on taxpayers. This paper tests a potential solution to this issue: an agricultural futures portfolio optimally selected to balance or transfer the cost (basis risk) of offering a hedge in the price of non-commodities like the premium Hass avocado. For this purpose, the present paper shows the results of Hass avocado price replication with this futures portfolio. The hypothesis tested herein is:

H0. 

Using a portfolio of agricultural futures traded in the CME or the NYMEX, optimally selected with the minimum tracking error method, leads to a proper replication of the premium Hass avocado mean price in Mexico.

It is essential to mention some issues related to the working hypothesis. First, agricultural futures trades are intended in the CME and NYMEX because these are the most liquid derivatives exchanges in the world [34]. Also, these agricultural futures are among the most traded in the U.S. markets [35]. The use of other agricultural and non-agricultural futures (like dairies), or even testing with futures of other non-U.S. exchanges, is left for further research. The core idea is to perform the first minimum tracking error portfolio test to replicate a non-commodity price like that of the Hass avocado.

The authors decided to use the minimum tracking error method for optimal futures portfolio selection because it is a straightforward method related to asset–liability management portfolio selection [21,22,23,25,36]. For this paper, and as mentioned in the previous section, the avocado price will be the “liability” or benchmark to replicate, reducing the basis risk or difference between the futures portfolio’s return and this benchmark.

Other relevant methods use artificial intelligence (a topic of general interest when writing this paper). The papers of Leung et al. [37], Mari and Mari [38], and Liao et al. [39] suggest using neural networks or genetic algorithms in reinforced learning applications for optimal portfolio selection, non-linear methods that complement the classical tracking-error mean-variance selection process. Because this paper is one of the first tests of using agricultural futures portfolios for non-commodity price hedging, the authors set aside these models and left them for further research.

The author’s central position is that if the simulated portfolios properly replicate the avocado price, they could be used for hedging positions.

When writing these lines, no works have tested or suggested alternative income or price hedging policies in Mexican non-commodity agricultural products. Similarly, no works deal with this issue in other regions or countries (like the EU, the U.S., or commodity exporters). This paper is among the first to test an alternative price and income reduction method with a lower impact on taxpayers, with a non-commodity price replication portfolio of agricultural futures.

Also, this paper is the first in the Mexican economy to suggest and test a hedging strategy in a non-commodity agricultural commodity: supreme-quality Hass avocado.

Several works have tested the benefit of using futures hedging strategies in their corresponding spot market commodity prices. It is not in the interest of this paper to review them all. The ones discussed herein are among the most related to this paper’s test and the ones that motivate the tests and results depicted herein.

This paper’s leading theoretical and practical motivations having been discussed, the following section depicts the data gathering and processing method. Briefly, it discusses the minimum tracking error optimal selection simulated weekly. The hedging methods tested are also briefly explained.

3. Materials and Methods

Given this paper’s theoretical motivations, it is worth mentioning how the authors gathered the data and made the simulations for avocado price hedging. For such simulations, the following assumptions are established:

• The hedged avocado production was sold over a one-week ($t + 1$), four-week ($t + 4$), and three-month ($t + 12$) period.
• A public or private agent (or financial intermediaries) is interested in offering an ask–settle price for avocado production (offering a price hedge).
• This agent balances the offered hedge by buying an optimal (minimum tracking error) portfolio of the following agricultural futures traded in the CME and NYMEX:
  • The 1-month expiration cocoa future traded in CME.
  • The 1-month expiration coffee future traded in NYMEX.
  • The 1-month expiration yellow corn future traded in CME.
  • The 1-month expiration wheat future traded in CME.
  • The 1-month expiration rough rice future traded in CME.
  • The 1-month expiration soybean future traded in CME.
  • The 1-month expiration sugar future traded in NYMEX.

The previous futures contracts were selected due to their agricultural nature and because these seven agricultural future contracts are among the most traded in the U.S. futures markets [35].

To estimate the optimal portfolio with the lowest tracking error, the tracking error was estimated as in (1), using the weekly percentage return of the average price of the Mexican supreme quality Hass avocado (Hass avocado or avocado henceforth). This average weekly price was estimated from all the recorded prices in the main spot markets in Mexico. These prices come from the National Markets Information System (SNIIM for its acronym in Spanish) of the Mexican Economics Secretary. These prices are recorded directly from the main public markets of the main cities in Mexico at $t$. The Mexican average Hass avocado price ($P_{avo,t}$) was estimated at $t$ with the arithmetic mean of all these recorded prices. With this average price, the historical continuous price return was calculated as follows:

$$r_{bm,t} = \ln \left(P_{avo,t}\right) - \ln \left(P_{avo,t-1}\right)$$

(2)

A similar method was used to estimate the weekly continuous-time price return of the seven agricultural futures of interest:

$$r_{i,t} = \ln \left(P_{i,t}\right) - \ln \left(P_{i,t-1}\right)$$

(3)

The historical price data of the seven futures comprise the close settle price at $t$ from the CME and NYMEX databases through the Refinitiv platform.

The historical data of the mean avocado price $P_{avo,t}$ and each future of interest $P_{i,t}$, ran from 9 January 1998 ($t_0$) to 29 September 2023.

To perform the simulations of interest, the weekly historical return data were used from $t_0$ to the simulated week ($t$) in a simulations time range from 1 January 2000 to 29 September 2023. The historical datum used to perform the optimal portfolio selection was an updated vector variable from $t_0$ to the simulated date $t$:

$$X = \left[{\mathbf{T}\mathbf{E}}_{\mathbf{i},\mathbf{t}}\right]$$

(4)

The previous vector variable (matrix), ${\mathbf{T}\mathbf{E}}_{\mathbf{i},\mathbf{t}}$, is the tracking error or price return of the i-th future concerning the price return of the avocado price:

$${\mathbf{T}\mathbf{E}}_{\mathbf{i},\mathbf{t}} = \left[r_{i,t} - r_{bm,t}\right]$$

(5)

With this vector variable in (4), an expected tracking error vector ($\mathbf{e}$) was estimated with the arithmetic mean values of the returns of each futures tracking error:

$$\mathbf{e} = \left[\overline{{\mathbf{T}\mathbf{E}}_{\mathbf{i},\mathbf{t}}}\right]$$

(6)

Also, a time-fixed tracking error variance–covariance matrix was estimated with $X$:

$$\mathsf{\Sigma } = {\left[{\mathbf{X}}^{\mathbf{\prime }}\mathbf{X}\right]}^{n - 1}, \mathbf{X}={[\mathbf{T}\mathbf{E}}_{\mathbf{i},\mathbf{t}}]-\mathbf{e}$$

(7)

With the expected tracking error vector and the covariance matrix in (6) and (7), the following quadratic programming problem was solved to select the optimal futures position (weights) $\mathbf{w}\mathbf{*}$:

$$\underset{\mathbf{W}}{\arg \min }\mathbf{w}\mathsf{\Sigma }\mathbf{w}$$

(8)

subject to:

(1)

${\mathbf{w}}^{\prime }1 = 1$

(2)

$\mathbf{w} \ge 0$

This optimal weight vector ($\mathbf{w}=\left[{\mathbf{w}}_{\mathbf{i}}^{\mathbf{*}}\right]$) gives the investment level that each future must have in the balancing position of the avocado price replication portfolio, leading to an estimate of the simulated portfolio’s return at $t$.

$$r_{p,t} = \sum {\mathbf{w}}_{\mathbf{i}}^{\mathbf{*}}\cdot r_{i,t}$$

(9)

To test the working hypothesis in this paper, 127 combinations of futures were used to select the optimal portfolio at $t$ with (8). These combinations range from sets of one to seven of the futures of interest. The core idea is to test the benefit of using different futures combinations to determine the most appropriate for avocado price replication.

To test such replication benefits, it is necessary to select the futures combinations and their optimal investment levels that lead to the most significant reduction of the tracking error in (1), that is, reaching values of ${TE}_{p,t}\approx 0$. To determine such effectiveness, the best-replicating portfolio must have the highest hedging effectiveness, estimated as follows:

$$H{E}_{p,t} = 1-{\displaystyle \frac{\mathsf{\sigma }{\left(r_{avo,t} - r_{p,t}\right)}^2}{\mathsf{\sigma }{\left(r_{avo,t}\right)}^2}}$$

(10)

The simulated futures portfolio with the $H{E}_{p,t}$ value closest to 1 is the one that replicates the avocado price the best. It is the most suitable combination to use as the balancing position of the avocado price hedge ($r_{avo,t} - r_{p,t}$). The hedging effectiveness in (10) assumes a “naïve” hedging strategy. That is, it assumes a 1-to-1 short futures position as spot one. Following Ederington [40], Ederington and Lee [41], Myers and Thompson [42], and Martinez and Zering, there is an “optimal” hedging position that does not necessarily use the 1-to-1 short futures position. There is an optimal hedging ratio that leads to better hedges and better hedging effectiveness. This hedge ratio $\mathsf{\beta }$ could be estimated with a least square regression model, with the futures return as a regressor and the spot position (avocado in the case of this paper):

$$r_{avo,t} = \mathsf{\alpha } + \mathsf{\beta }\cdot r_{p,t} + {\mathsf{\epsilon }}_t$$

(11)

Given the optimal hedge ratio in (11), the optimal hedging strategy could be simulated as ($r_{avo,t}-\mathsf{\beta }{r}_{p,t}$). This strategy assumes that a 1-to-1 hedging position is not necessary. A 1-to-$\mathsf{\beta }$ could lead to better hedging and basis risk reduction.

Using either a 1-to-1 or a 1-to-$\mathsf{\beta }$ hedging position assumes that the same position or weight in the futures portfolio is time-fixed. This implies a passive hedging strategy that assumes that an avocado producer will hedge each time she or he wants to sell avocados.

To relax this assumption, the authors used a simple active hedging strategy with Markov switching (MS) models [43,44,45]. For this specific case, the avocado’s return could be modeled in a Gaussian two-regime context:

$$r_{avo,t} \sim \mathsf{\Phi }\left({\overline{r}}_{avo,s,t},{\mathsf{\sigma }}_{avo,s,t}^2\right)$$

(12)

The first regime $(s = 1$) corresponds to “calm” or “normal” time periods with low volatility returns and, the second ($s = 2$), to “distress” periods with high volatility. This implies ${\mathsf{\sigma }}_{avo,s=1,t}^2 < {\mathsf{\sigma }}_{avo,s=2,t}^2$ and ${\overline{r}}_{avo,s=1,t} > {\overline{r}}_{avo,s=2,t}$.

With the use of MS models, two trading rules were simulated:

• To perform the naïve hedging rule if the forecasted probability of the high volatility regime is high (${\mathsf{\xi }}_{s=2,t+1} > 50\%$).
• To perform the optimal hedging ratio rule ($r_{avo,t} - {\mathsf{\beta }}_{\mathrm{s}=2}\cdot r_{p,t}$), also if ${\mathsf{\xi }}_{s=2,t+1} > 50\%$).

Consequently, this paper will test four hedging strategies: the single-regime naïve and optimal hedge ratio hedging strategy, and using one of these two trading rules only if the forecasted probability (${\mathsf{\xi }}_{s=2,t+1}$) of high volatility periods is higher than 50%.

MS models filter the regime or state of nature from which a given realization in a time series ($r_{avo,t}$) is derived. They assume that each realization is generated through an ergodic Markovian chain (a two-regime one in the case of this paper) with a $2\times 2$ transition matrix $\mathsf{\Pi }$, and smoothed probabilities of each realization being in the s-th regime at $t\left({\mathsf{\xi }}_{\mathrm{s},\mathrm{t}}\right)$. As mentioned, MS models filter the parameter vector from the data through Bayesian machine learning methods (the EM algorithm [46] or Markov chain Monte Carlo simulations [47,48,49]):

$$\mathsf{\theta } = \left[\mathsf{\Pi },{\mathsf{\xi }}_{\mathrm{s},\mathrm{t}},{\overline{r}}_{avo,s,t},{\mathsf{\sigma }}_{avo,s,t}^2\right]$$

(13)

In the previous expression, $\mathsf{\Pi }$ has the regime-specific transition probabilities ${\mathsf{\pi }}_{i,j}$ from regime $i$ at $t - 1$ to regime $j$ at $t$.

The practical use of MS models could be (1) to filter and determine the probability of being at each regime at $t$ to characterize the time series of interest or (2) to use these smoothed probabilities for trading or business decisions. The latter use is one of interest for this paper. The first proposal for using the MS model for trading decisions comes from Brooks and Persand [50]. These authors used Gaussian two-regime MS models to invest in UK, U.S., and German stock with an investment level given by ${\xi }_{s=2,t}$ and in government bonds in an investment level given by ${\xi }_{s=1,t}$. These authors outperformed a buy-and-hold strategy either in the stock indexes of each country or in a portfolio invested in the corresponding government bonds. Following Brooks and Persand, other authors tested the use of MS models or even MS models with time-varying variances (generalized autoregressive conditional heteroskedastic variances), also known as MS-GARCH. Some tests were made in stocks and indexes or active portfolio management [51,52,53,54,55,56]. The main findings of these works are that using MS or MS-GARCH models to shift the investment levels of the managed portfolio led to better performance than a buy-and-hold strategy. For the specific use of commodities (agricultural among these) trading, several works tested the use of MS and MS-GARCH models for trading decisions [57,58,59,60,61,62,63]. These works show better high-volatility regime forecasts for trading or hedging, leading to better results in active futures trading in corn, coffee, cocoa, wheat, soybean, and sugar. Works such as the one of Lien et al. [60] used MS-GARCH hedge ratios (${\beta }_s$) and found better hedging results in agricultural commodities futures, a result that the authors wish to extend in the present paper.

Following the previous methodology, the hedging effectiveness $H{E}_{p,t}$will be the key parameter to determine whether one of the 127 combinations of futures portfolios is the best option to replicate the performance of the avocado price and, consequently, to be used as a potential balancing position of a hedge issuance for avocado producers. This parameter will be tested in the four previous scenarios summarized as follows:

• A naïve hedge of the avocado price return ($r_{avo,t}$) with a 1-to-1 short position in the simulated futures portfolio.
• A hedging position given by the optimal hedging ratio ($\mathsf{\beta }$) in a single-regime scenario.
• A naïve hedge of the avocado price return if ${\xi }_{s=1,t+1}>50\%$.
• A dynamic hedging position with a regime-specific hedging ratio (${\beta }_{s=2}$), if ${\xi }_{s=1,t+1}>50\%$.

The authors expected that a more diversified futures portfolio (preferent with the seven futures of interest) would be the one with the highest (closest to 1) $H{E}_{p,t}$. Therefore, the selection criterion of the best combination for hedging purposes will be its rendering the highest $H{E}_{p,t}$.

The authors programmed the weekly simulations of estimating the optimal minimum tracking error portfolio in R scripts, using SQLite databases to store the results and the fPortfolio [64], Quantmod [65], and MSwM [66] libraries for optimal (minimum tracking error) portfolio selection and Gaussian time-fixed variance MS models estimation with the EM algorithm.

In this first review for non-commodity agricultural price hedging, it will be of interest to use the seminal Gaussian two-regime MS model in the high-volatility regime ($s = 2$) forecast at $t + 1$ with the following estimation method, given ${\mathsf{\theta }}_{\mathbf{t}}$ in (13) at t:

$$\left[\begin{array}{c}{\xi }_{s=1,t+1}\\ {\xi }_{s=2,t+1}\end{array}\right] = \mathsf{\Pi }\left[\begin{array}{c}{\xi }_{s=1,t+1}\\ {\xi }_{s=2,t+1}\end{array}\right]$$

(14)

The simulation method and data gathering and processing having been detailed, the next section discusses the main results and findings.

4. Results

Basis risk is among the main drawbacks of using agricultural commodity futures to hedge non-commodities. To offer a general idea, Figure 2 shows the historical performance of the base 100 (since 1 January 2000) Mexican mean Hass avocado price, along with the historical values of the seven agricultural futures of interest.

As noted, the avocado price is increasing due to the growing demand from U.S. consumers. Contrary to the seven agricultural commodities, this non-commodity has a lower output, and the demand has been stronger since 2000 (when the U.S. borders were opened to this product’s imports). This general behavior makes the avocado price less cointegrated with these futures, showing price increases in periods such as 2008 and 2020, increases that revert to lower (perhaps equilibrium) levels in posterior periods. Consequently, using one of these futures to hedge the avocado price is useless.

Following this result, 127 combinations (portfolio simulations) were performed, starting with seven single-future portfolios to a set of a portfolio with seven futures (the ones of interest herein). Appendix A shows the results of the simulations.

Table 1. The simulated portfolios with the best hedging effectiveness in the four scenarios and three hedging horizons of interest.

Hedging Strategy	Hedging Horizon	Simulated Portfolio	Hedging Effectiveness	Optimal Hedging Ratio ($\mathsf{\beta }$)
Single-regime naïve	t + 1	Sugar–Coffee	0.9387	1.0000
Single-regime naïve	t + 4	Sugar	−0.3517	1.0000
Single-regime naïve	t + 12	Wheat–Rough Rice–Sugar–Coffee–Cocoa	−1.0960	1.0000
Single-regime optimal hedging ratio	t + 1	Sugar–Coffee	0.9434	1.0762
Single-regime optimal hedging ratio	t + 4	Wheat–Rough Rice–Sugar–Coffee–Cocoa	0.0000	0.9067
Single-regime optimal hedging ratio	t + 12	Wheat–Rough Rice–Sugar–Coffee–Cocoa	0.0000	1.1058
Two-regime naïve	t + 1	Corn–Rough Rice	0.4888	1
Two-regime naïve	t + 4	Corn–Rough Rice	0.4888	1
Two-regime naïve	t + 12	Corn–Rough Rice	0.4888	1
Two-regime optimal hedging ratio	t + 1	Wheat–Cocoa	0.4445	1.1329
Two-regime optimal hedging ratio	t + 4	Wheat–Cocoa	0.4434	1.0959
Two-regime optimal hedging ratio	t + 12	Wheat–Cocoa	0.4445	1.1096

summarizes these results by showing the portfolios with the best hedging effectiveness (10) in each of the four scenarios of interest (single- or two-regime).

As noted, the single-regime hedging strategies show an important income risk reduction in the avocado price over the $t+1$ hedging horizon. Using longer time horizons does not lead to a significant income risk reduction. In some cases, it is worst to use the hedging strategy, as in the case of a naïve strategy in a single regime for $t+4$ and $t+12$ hedging horizons.

The last column of Table 1 shows the optimal hedging ratio used in each scenario. The naïve strategy rows show the 1 beta value assumed in this strategy and the optimal hedging ratio, the beta values either in the single-regime context or the high-volatility beta value used for hedging.

Of all 127 futures combinations (portfolios) tested herein, a portfolio with sugar and coffee futures is the best option to hedge the avocado price in a single-regime context. This conclusion is due to the 0.94 hedging effectiveness that this portfolio shows. That is, the hedging effectiveness is close to the ideal value of 1, suggesting a proper income risk reduction.

In the two-regime context, the best-performing (best-hedging) portfolios are those with corn and rough rice futures in a two-regime naïve strategy and the wheat–cocoa portfolio in the optimal-hedging-ratio one. Despite this, using a futures portfolio to hedge (to replicate) the avocado price in a two-regime only reduces the income risk by 44–48%, leading to the conclusion that the naïve strategy in a single-regime hedging context is preferable by using a sugar–coffee futures portfolio to replicate the Hass avocado price in a $t+1$ hedging horizon.

It is important to highlight that using a sugar–coffee portfolio leads to similar hedging effectiveness results. It is preferable to use the naïve strategy because the optimal hedging ratio of the same portfolio is close to 1. Consequently, the avocado price could be replicated (and hedged) with this naïve hedging strategy to reduce the price or income risk.

To show the causes of this hedging effectiveness in the sugar–coffee futures portfolio, Figure 3 shows the historical performance of the simulated portfolios, depicted in Table 1, against the base 100 value of the avocado price at $t$.

As noted in Figure 3, the sugar–coffee and wheat–cocoa portfolios have a simulated value close to the avocado price. The main difference comes in the 2008–2021 period, in which the sugar–coffee portfolio (as in the whole time series) has the closest fit to the non-commodity of interest. This result comes from the mean investment level in sugar and coffee futures (83.48% and 16.52%, respectively), depicted in Figure 4 and

Table 2. The mean future investment level in the simulated portfolios with the best hedging effectiveness in the four scenarios and three hedging horizons of interest.

Simulated Portfolio	Mean Investment Level (%) in Each Future
	Cocoa	Coffee	Corn	Rough Rice	Soy Bean	Sugar	Wheat
Sugar–Coffee		16.5166				83.4834
Corn–Rough Rice			59.7291	40.2709
Wheat–Cocoa	19.2849						80.7151
Wheat–Rough Rice–Sugar–Coffee–Cocoa	3.4962	3.5369		14.0753		38.7131	40.1785
All futures	7.1427	7.0626	41.601	5.7315	9.3856	17.1066	11.9699

.

From the results in Table 2 and Figure 3, it is interesting to highlight that even if Table 1 shows that a single future portfolio leads to poor hedging results in a non-commodity such as avocados, a linear combination or portfolio of sugar and coffee futures enhances avocado price replication. Also, for the specific case of the mean national Hass avocado price in Mexico, the futures of the NYMEX (coffee and sugar) are the most suitable for hedging this non-commodity.

Following this portfolio’s results, it is interesting to test whether the relationship between the avocado price and the sugar–coffee portfolio is long-term.

Table 3. Unit root and Engle–Granger cointegration test of the simulated portfolio with the avocado’s price.

Simulated Portfolio	Unit Root Test (p-Value)	Cointegrating Relationship ($\mathsf{\alpha }$)	Cointegrating Relationship ($\mathsf{\beta }$)	Cointegration Test (p-Value)
Corn–Rough Rice	0.0495	58.7429	0.5583	0.0100
Sugar–Coffee	0.0814	1.3102	0.9173	0.2969
Wheat–Cocoa	0.0975	29.3667	0.7466	0.0100
Wheat–Rough Rice–Sugar–Coffee–Cocoa	0.0812	53.1511	0.4629	0.0100
Avocado	0.0688	-	-	-

shows the augmented Dickey–Fuller [67,68] unit root test (with one lag in the residuals, no time trend, and with drift) and the Engle–Granger [69] cointegration test.

This table shows that the four simulated portfolios and the avocado time series are not stationary at the 10% significance level or lower. The second and third columns show the alpha and beta values of the cointegrating relationship, and the fourth column shows the p-value of the unit root test of the residuals of such relationship (Engle–Granger test). As noted, only the sugar–coffee portfolio shows a long-term relationship with the avocado’s price. Therefore, Table 3 gives more proof that using this portfolio is preferable to the other three portfolios, because it not only enhances the hedging effectiveness in the avocado’s price but also its value has a long-term relationship with this non-commodity.

5. Discussion

The results from Table 1, Table 2 and Table 3 show that, as theoretically expected, using single agricultural commodity futures is not useful to hedge non-commodities like the Hass avocado. The main result is that using an agricultural futures portfolio leads to a proper avocado price replication for hedging purposes. From all the combinations of portfolios with the seven futures of interest herein (coffee, corn, rough rice, soybean, sugar, and wheat), a portfolio made of a mean investment level of 83.48% in sugar futures and 16.52% in coffee leads to a very close price replication, one with hedging effectiveness of 94% of the avocado’s price fluctuations.

As mentioned in the introduction, the idea behind replicating the avocado price is to have a balancing position to offer a hedge of the avocado price. Because there are no derivatives to hedge the price of this non-commodity, using a synthetic hedge through a futures portfolio could be an answer. A given public (government) or financial institution in Mexico could offer to hedge the avocado price and buy a sugar–coffee futures portfolio to cover or balance the risk exposure for such issuance. Because this portfolio can replicate the avocado price with a proper fit and a given institution could offer such hedging, the avocado producers could have a hedging effectiveness of 93–94%. This means that they could hedge almost entirely a downward move of the avocado price in a $t+1$ hedging horizon.

Despite this interesting result, to hedge the avocado price with the sugar–coffee futures portfolio leads to poorer hedging effectiveness in hedging horizons of $t+4$ and $t+12$ weeks.

As a corollary of the results discussed herein, using a synthetic hedge with an agricultural futures portfolio of the avocado price is feasible It could help cover avocado producers’ income risk (the author’s working hypothesis is proved as true partially).

6. Conclusions

Hedging the price of non-commodities with commodities futures is an activity that needs further review to translate or share the risk of commodity price changes. The main issue with this practice is the presence of basis risk, that is, the future difference between the non-commodity’s spot price and the futures position. This result could reduce or even increase the price risk in the best-case scenario.

Nonetheless, this paper tests a first method to hedge the price and, consequently, the income of premium-quality Hass avocado producers. This non-commodity is an agricultural sector that is raising demand around the world. Its popularity as a healthy food and fruit used in “haute cuisine” and as the fundamental ingredient of dips and appetizers has motivated important price activity and volatility.

This fruit is among the main agricultural exports in countries like Mexico, Chile, Colombia, Perú, and Israel (among others). Even apart from this, it is interesting to test avocado price hedging methods, especially in the Mexican market, which is the main producer of this fruit worldwide and creates economic value through its related activities.

To hedge such prices, this paper tested using the seven main agricultural futures traded in the Chicago Mercantile Exchange (CME) and the New York Mercantile Exchange (NYMEX): corn, cocoa, coffee, rough rice, soybean, sugar, and wheat. Because these futures prices and the avocado are not cointegrated and show a high level of basis risk in a hedging strategy, the authors tested the use of a futures portfolio, optimally selected through the minimum tracking error method, method in which the hedging position $\mathbf{w}\mathbf{*}$ is optimally selected by minimizing the difference between the portfolio’s percentage change and the benchmark’s (the Mexican avocado spot price in this test).

The core idea is to find such optimal investment levels $\mathbf{w}\mathbf{*}$ that will reduce the basis risk to zero. The hedging effectiveness $H{E}_{p,t}$ is a metric used to measure how effective a hedging (futures or options) position is in reducing basis risk. A value of $H{E}_{p,t}\approx 1$ suggest a perfect fit (zero basis risk) between the hedging position and the spot one. A value $H{E}_{p,t}\approx 0$ suggest a poor hedge, and a value of $H{E}_{p,t}<0$ shows that the basis risk increases. The hedging position adds more price fluctuation than the single or unhedged spot one.

This paper tested the optimal selection of the futures portfolio to replicate the avocado price in a hedging horizon of $t+1$, $t+4$ and $t+12$ weeks from 1 January 2000 to 29 September 2023. The optimal portfolio selection was tested in 127 different portfolios, each combining the seven futures of interest. The core idea was to find the best-fitting optimal futures combination to replicate the avocado’s price.

The goal of properly replicating the avocado price to reach $H{E}_{p,t}$ values near to 1 show an important avocado price hedging method that could help a Mexican public or financial institution (like SEGALMEX, a Mexican public company that hedges agricultural products and seeks Mexican food security) to offer an avocado price hedging for avocado producers. Offering such a hedge could translate the price risk to this institution, a risk that could be balanced by buying the optimally selected futures and portfolios tested herein.

The results of the simulations show that using an optimal portfolio with a mean investment level of 16.52% in coffee and 83.48% in sugar futures will lead to a hedging effectiveness of 0.94. Consequently, a Mexican public or financial institution could buy this simulated portfolio to balance the price hedge offered to Mexican avocado producers.

It is essential to highlight that this result holds in a $t+1$ hedging scenario and the $t+4$ and $t+12$ ones need further review or testing of other optimal selection methods.

One of the drawbacks of the simulations tested herein is that the optimal portfolio selection was made through a minimum tracking error optimal portfolio selection. Even if this method is appropriate, some alternative methods exist, such as cointegration [70,71]. These are less computationally intensive and in line with multivariate time series analysis due to potentially long-term relationships between variables. Such a technique is left for further research, and this paper’s method is among the first of several tests intended for non-commodities price hedging.

Another drawback is that the synthetic hedge could be tested with a better active hedging strategy. The authors used Markov switching (MS) models in this paper to forecast the probability of high-volatility regimes. If this value was high, the algorithm made either a naïve or optimal hedge ratio hedge. Among the drawbacks of such an active hedging method is using time-fixed variances in the MS models and time-varying GARCH ones. Because the time-fixed variance changes over time and is slower to converge when the time series is in the low-volatility regime, MS-GARCH models could lead to better high-volatility regime forecasts. Consequently, these could lead to a better hedging effectiveness than the average 44% shown in the wheat–cocoa portfolio of Table 1. Therefore, using MS-GARCH models could be an exciting extension of these results, and further research is also needed.

Another limitation of these first results is that the optimal portfolio selection was made with time-fixed covariances. Consequently, the covariance (correlation) was supposed to be fixed through time. The use of GARCH covariances [72,73,74] or even MS or MS-GARCH ones [75] could be of interest to forecast the covariance matrix in a single- or two-regime context, leading to better optimal portfolio selection processes and, potentially, to improve the result in the $t+4$ and $t+12$ hedging horizons.

Also, these GARCH and multiple-regime covariances could be estimated with exogenous factors such as news or social media sentiment indexes and financial or economic factors to enhance the information set and create better-fit estimations of such covariances.

Using agricultural futures from the CME, NYMEX, and other non-agricultural or financial commodities in the optimal hedging futures portfolio is a natural extension of this paper’s results. It could lead to further improvements in hedging horizons beyond $t+1$.

The growing influence of cryptocurrencies on financial markets, as Li et al. [76] suggest, is another factor that should be estimated in future extensions of this paper’s test.

Finally, using logit models [53,54] or even neural networks to forecast the regime shift for active hedging decisions could be considered.

This work as one of the first to test the use of quantitative optimal futures portfolio selection to hedge the price of non-commodities with futures portfolios, using the case of the Mexican avocado, which could be extended to other agricultural products in Mexico and abroad.