3.1 Target returns

We assume that a risk-averse forage producer would prefer the least risky option among several alternatives that yield the same expected return. By participating in the Annual Forage, producers can protect a portion of the value of their crop while reducing the variability of their returns. Consider the net farm sales (π), given by the production and sale of forage, which fluctuate with the precipitation received. For a given price and cost of production, drier years are expected to be associated with lower returns and vice versa.² Therefore, to reduce the variability of returns, a forage producer can offset this risk by ensuring through the Annual Forage program the expected value of the loss in yield.

We further assume that the target expected return is equal to the expected value of forage under favorable precipitation conditions (𝑣). Hence, to achieve this target return, the producer should select a coverage that yields an expected net indemnity equal to the expected loss in net farm sales caused by limited precipitation. Namely, the expected net farm income should be equal to:

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where E(π) are the expected net farm sales, under all possible precipitation scenarios, and E(R) is the corresponding expected value of the net indemnity needed to stabilize the net farm income around 𝑣.

Again, the goal of a risk-averse producer is to choose a coverage with an expected net indemnity of E(R) that minimizes the variance of the net farm income, which is given by

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where is the variance of the net farm sales, σ_jk is the covariance between r_j and r_k, and σ_πj is the covariance between π and r_j.

3.2 Portfolio-based strategies

Portfolio optimization methods have been used to develop risk-efficient coverages for rainfall index-based insurances such as the Pasture, Rangeland Forage (PRF) program (Zapata and García, 2022; Zapata et al., 2024). In this study, we adapted and evaluated different coverage selection strategies to the specifications of the Annual Forage insurance program. Compared to Zapata et al. (2024), we incorporated net farm sales into the selection process, presenting a more realistic application of robust portfolio analysis. Overall, the objective of portfolio-based coverage selection strategies is to obtain an expected net farm income at the lowest risk possible, where risk is commonly measured by the corresponding variance or standard deviation of the coverage’s returns.

The mean-variance (MV) approach is a common specification of the coverage selection problem (Diersen et al., 2015; Jimenez Maldonado, 2011; Zapata and García, 2022; Zapata et al., 2024). It is an intuitive strategy that relies on observed sample moments to identify optimal coverage, but might underperform in real out-of-sample circumstances. However, the MV selection strategy can be adjusted to improve its out-of-sample performance. For this aim, we considered two general alternative specifications based on shrinkage and ensemble learning techniques. The shrinkage approach represents a penalized selection of the coverage parameters to produce a more stable coverage. On the other hand, ensemble procedures are based on random subspace methods to generate and combine multiple coverages for a more robust selection.

Mean-variance approach

The objective of the MV coverage selection strategy is to identify a coverage that minimizes the variance of net farm income subject to achieving the target return. Namely, the MV coverage selection problem can be represented as (Zapata and García, 2022):

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where w is a vector containing the distribution of the percent of value (w₁, w₂, … , w₆), and d is the corresponding vector of indicator variables, such that d_j equals one if the jth index interval is selected (i.e., w_j > 0 ), and zero otherwise. The above V(Π) and E(Π) are computed according to equations 5 and 4, respectively, using the historical rainfall index values, time-adjusted forage yields, and current prices, costs, and actuarial information. Particularly, for each recorded year, the net farm income corresponding to the selected coverage parameters is calculated. These annual returns are then used to compute the sample moments.

Random subspace methods

Coverage selection strategies based on random subspace methods generate an assembled coverage by combining multiple coverage alternatives. Two ensemble strategies are adapted to the context of the Annual Forage program, bagging selection (BS) and random subset selection (RSS). BS uses bootstrapping techniques to create random samples in which optimal coverages are identified and then compounded into a single selection (Zapata et al., 2024). Like the BS strategy, RSS generates aleatory subsets, but the search space for the optimal coverage at each subsample is also a random subset of all possible combinations of the selection parameters (Zapata et al., 2024).

The assembled coverages derived from diverse selections are expected to produce more stable returns. To obtain heterogeneous coverage options, random subsamples are created by bootstrapping the historical rainfall indexes and the set of valid combinations of coverage level, productivity factor, and index intervals. Past rainfall indexes are represented by the set 𝕀 = {𝕀₁, 𝕀₂, … , 𝕀_N}, where 𝕀_i = {I_1i, I_2i, … , I_6i} are the rainfall indexes corresponding to the ith year. Similarly, there is a finite set ℂ = {C₁, C₂, … , C_M} containing the M feasible combinations of the decision parameters, where C_m = {λ_m, γ_m, d_m} represents the mth combination of coverage level, productivity factor, and index intervals.

Optimal coverage of the BS strategy is derived by sampling the set 𝕀, while both 𝕀 and ℂ sets are sampled in the RSS strategy. In BS, B bootstrapped samples (𝕀^b, b = 1, … , B) of size N are generated with replacement from the set of historical rainfall indexes (𝕀). A characteristic of the rainfall index is that it has a mean of approximately 100 for each of the J index intervals (i.e., ). To preserve this property, the sampled rainfall indexes of the 𝕀^b bootstrap sample are recalibrated to have a mean of 100. Namely, the adjusted mth realization of the rainfall index of the jth index interval in the bootstrap sample b is:

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where is the average of the sampled rainfall indexes from the jth index interval at the bth bootstrap iteration. Once the rainfall indexes are calibrated, optimal coverage is derived using the MV approach described above for each bootstrap sample considering all the valid combinations of the selection parameters in ℂ. The RSS differs from the BS approach in that the former evaluates a random subset of 𝜚 elements of ℂ to find the optimal coverage for each bootstrap sample (w*^b), where 1 ≤ 𝜚 ≤ M. Therefore, the resulting B coverages are less correlated with each other. The appropriate number of bootstrap samples (B) and relative size of the search space ( ) are also estimated using cross-validation techniques.

The individual coverage selections from the bootstrap samples (w^*b, b = 1, … , B) are then combined to derive an assembled coverage. This merged coverage must comply with the Annual Forage coverage selection restrictions. To this aim, the following aggregation rule is used for both the BS and RSS strategies. Selected coverage levels and productivity factors from the B bootstrap samples are combined by computing their means, which are rounded to the nearest valid levels. In the case of the distribution of the percent of value, the individual sample allocations are aggregated in a manner that minimizes the sum of squared differences between the percent of value of the bootstrap coverages and the assembled coverage, subject to satisfying all allocation restrictions. Namely, the random subspace allocation of the percent of value (w^bag) is the solution to

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where is the optimal percent of value for the jth index interval in the bth bootstrap sample.

4.1 Production and market data

Texas is the largest cattle producer in the USA (USDA NASS, 2024). However, data on the state’s forage production is scarce. Due to the lack of publicly available historical records at the farm or county levels, the proposed coverage selection strategies were illustrated using cool-season forage variety trial results. Forage wheat yield data were compiled from Texas A&M AgriLife’s cool-season annual forage variety trial reports for the South Texas demonstration site (i.e., College Station) (Texas A&M AgriLife, 2022). Variety trial data are available from 2010 to 2021, and the recorded yields (dry ton/ac) are presented in Figure 1. Forage yields were estimated by clipping and weighing the production of each plot during the growing season. Forage wheat varieties were planted in late September and early October each year and were grown under similar irrigation and fertilization regimes. Different varieties were planted over the years, but all exhibited the same yield trends. Therefore, we selected a set of varieties that remained relatively constant throughout the evaluation period. The observed forage wheat yields were used to project forage yields for the period of available rainfall indexes (i.e., 1949–2024). Therefore, the predicted yields represent potential forage yields of existing forage wheat varieties under past growing conditions.

In consultation with local forage specialists, forage yield was assumed to be a function of minimum temperatures and drought conditions during the growing season, which runs from late September to March. To estimate a functional form for the conditional expected yield, observed forage yields were regressed on the corresponding minimum temperatures and Palmer Z index (a monthly measure of short-term drought) for Brazos County reported by NOAA NCEI (2024). To reduce the correlation between the explanatory variables, both monthly minimum temperatures and the Palmer Z index were grouped into periods with similar effects (i.e., October, November to December, and January to March).

Ordinary Least Squares (OLS), Ridge and Lasso regression methods were evaluated to identify the model specification with the best out-of-sample performance in predicting yield. Models’ performance was measured by the mean squared error³ (MSE). An 80/20 train-test split was used, with 80% of the data used to fit each yield model and the remaining 20% used to test it. The train-test process was repeated 1000 times using different random samples to reduce possible sampling effects. The OLS approach yielded the lowest MSE among the three estimation alternatives and it was used to estimate the conditional expected yield function. The entire dataset of observed yields and weather conditions was then used to estimate and project forage wheat yields from 1949 to 2024. The estimated OLS yield regression parameters are presented in Table 1 and the estimated yields in Figure 1.

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Observed and estimated forage wheat yields used to calculate farm sales. CI, confidence interval.

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Estimated forage yield coefficients.

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Projected yields along with corresponding forage prices and cost of production were used to calculate the expected net farm sales. Namely, historical Texas hay prices were obtained from USDA NASS (2024) and the production costs were adapted from the Coastal Bend region dryland wheat enterprise budget developed by Texas A&M AgriLife Extension (2024). In 2024, the variable cost, which consists of cutting and baling the hay, was estimated at $72.92 per ton and all other production expenses totaled $202.84 per acre. The Price Paid Index for Commodities, Services, Interest, Taxes, and Wage Rates (PPITW) (USDA NASS, 2024) was used to generate cost estimates for the years 2017 through 2023, with 2024 production costs as baseline. Past hay prices and deflated costs were used as observed values during the training and evaluation of the coverage selection models and are presented in Table 2.

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Realized and expected hay yields, prices, costs, and returns, 2017–2024.

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4.2 Actuarial data

The different coverage selection strategies were evaluated using the actuarial data from Grid 12735 located in Brazos County, TX, USA, where the forage variety trials were conducted. Although our empirical application focused on a specific grid in Texas, our data-driven methods are valid and can be directly extended to all participating regions and grids nationwide. By focusing on a single location, we were able to conduct a detailed analysis of the optimal coverages and contrast the different solutions in terms of indemnity payments and related risk. However, given the vast and diverse agroecosystems covered by the Annual Forages program, the coverages identified for Grid 12735 may not be directly extrapolated to other locations.

The rainfall indexes (i.e., 1949–2024) for the year 2024 were downloaded from the USDA RMA (2024c) decision support tool. As mentioned above, the rainfall indexes are adjusted each year to incorporate the latest realizations of the index and reset their average values around 100. For example, the historical rainfall indexes from 1949 to 2023 reported in 2024 have a mean close to 100. Unfortunately, only the most recent historical values of the rainfall index (i.e., adjusted for 2023) are publicly available. We used a recursive rescaling process to recover the corresponding “to-date” historical rainfall indexes for years prior to 2023. Namely, the rainfall index of the jth index interval for year i adjusted to reference year Γ s:

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In this way we recovered the set of historical rainfall indexes adjusted to each past year of interest (i.e., Γ for years 2016 to 2022). As discussed in the following sections, these recalibrated rainfall indexes are used in the training and tuning of the coverage selection models.

Also, corresponding county base values, premium rates, and subsidy factors, as well as the actual realization of the rainfall index for years 2017 through 2024 were obtained from the USDA RMA (2024d) actuarial information repository. This actuarial information was used to evaluate the performance of the coverage selection strategies by comparing projected returns with actual realizations during the testing phase. It is worth mentioning that for the 2024 coverage, the index intervals available to producers changed depending on the planting date. Before 2024, four 6-month growing seasons covered by 6 index intervals were available. Currently, protection is available for the following 6 months after planting. Therefore, in our empirical application, where the forage is assumed to be planted in September and harvested in March, the first growing season (i.e., September to March) was selected for years prior to 2024 and the default available index intervals to choose for 2024 were October–November, November–December, December–January, January–February, February–March and March–April. The distribution of the rainfall indexes of interest of Grid 12735 are presented in Figure 2.

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Historical distribution of rainfall index by index interval for Grid 12375, 1949–2024.

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4.3 Return and loss expectations

As described above, we consider that farmers make their coverage selection decisions based on their expectations about future returns and weather-related losses. Therefore, their expectations about coming forage yields, prices, and cost of production are incorporated into the selection process. In the absence of complete historical data, a pragmatic approach was followed to derive farmers’ production and market expectations. First, we assumed that each year farmers would want to secure a return equivalent to the expected return under favorable growing conditions. In our study, this optimistic return is represented by the net farm sales associated with an expected yield above the mean, produced and sold at expected hay costs and prices, respectively. To achieve this target return, a farmer could protect the expected loss caused by limited precipitation (i.e., E(R) in equation 4) by participating in the Annual Forage program.

Policyholders make their coverage selection decisions before planting with the information available at that time. Therefore, to emulate the farmers’ decision process, expected future yields, prices, and costs were updated each year by incorporating new outcomes. For a given year, the expected yield represents the average of all previous yields, excluding the one for the year being covered. All past observations are considered as the estimated yields characterized potential realizations of current forage wheat varieties (Figure 1). An optimistic expected yield is also estimated to calculate the target return. This optimistic yield only considers those past yields that are higher than the expected yield. On the other hand, price expectations are represented by a forecasted hay price using a first-order differencing specification with 2 lags. Different price regression models were trained using historical data from 1949 to 2016 and the specification that best fit the data was selected using the Akaike information criteria. Like expected yield calculations, the price forecasting model considers all past observations prior to the year of interest. Lastly, cost expectations for the coming year are given by the most recent cost estimates at the time (i.e., previous year values). Expected hay price, yield and cost of production are also included in Table 2.

We further assumed that farmers’ forage yields and the market hay price and custom harvesting costs (i.e., cutting and baling) are independent. Therefore, the expected value of the net farm sales is given by:

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where p is the hay price, y is the forage yield, h is the custom harvesting cost rate, and c represents all other costs. Similarly, the target return is equal to E(π|y > ȳ) = 𝑣, where ȳ is the average yield. The target, expected, and observed net farm sales are presented in Table 2.

4.4 Estimation, Tuning and Testing of Coverage Selection Models

A recursive training set was used to estimate the different coverage selection models. Particularly, annual coverage selections were derived for years 2017 through 2024, the period in which the Annual Forage insurance program has been used successively at the location of the study (USDA RMA, 2024a). For each year under consideration, the optimal coverage to attain the target return was identified using the expected forage yield, hay price and costs of production for that year, as well as its corresponding county base value, premium rates, subsidy factor, and the adjusted rainfall indexes (i.e., I_ji,Γ from equation 9). For instance, to estimate the optimal coverage for 2017, the selection models were trained using 2017 production and market expectations, and actuarial data along with the 2016-adjusted rainfall index from 1949 to 2016. All models used the same input data, and their proposed coverages were updated each year to obtain the target return by incorporating new farmers’ expectations and actuarial information.

The tuning parameter (τ) in the PS strategy and the number of bootstrap samples (B) and the proportion of the search space considered ( ) in the random subspace methods were estimated using cross-validation. Two 10-year validation sets were used to tune the hyperparameters of the coverage selection strategies considered. The first validation set consisted of the years 2007 to 2016 and was used to find the optimal value of the hyperparameters used in the coverage choices for the years 2017 to 2023. The second validation set (i.e., 2014 to 2023) was used to calibrate the hyperparameters for 2024 coverage. These two validation sets were used to accommodate the allowable index intervals prior to 2024. Optimal coverage was obtained for each of the 10 years of the validation sets using a discrete array of potential parameter values and the recursive estimation approach described above for the training set. However, compared to the optimal coverage selections in the training set, the county base value, premium rates, and subsidy factor used for each year in the first and second validation sets corresponded to actuarial information for the year 2017 and 2024, respectively. The expected net farm sales for each year in the validation sets were estimated using the corresponding 2017 and 2024 expected hay prices and cost of production. In addition, the average target indemnity payment in the test set (i.e., $54.76/ac) was considered each year in the validation sets.

The optimal value of the hyperparameters was obtained by calculating the MSE between the observed and expected net farm income in the validation set for each τ and combination of B and . The hyperparameter values with the lowest MSE in the validation set were selected and used in the training set. One hundred equidistant values from 50 to 10,000 were evaluated to identify the optimal value of τ. Based on its performance on the validation sets, τ was set to 7,990 to derive coverage for the years 2017 to 2023, and equal to 7,688 to obtain the optimal coverage for 2024. Similarly, the twelve combinations of three possible bootstrap replicates (i.e., 100, 300 and 500) and four search space ratios (i.e., 1, 2/3, 1/2 and 1/3) were evaluated. From both validation sets, the optimal number of bootstrap samples for the BS strategy was set to 300. In the case of the RSS, a combination of 100 bootstrap samples and a search space ratio of 0.50 resulted in the lowest MSE on the first validation set (i.e., 2007–2016). Similarly, using the second validation set (i.e., 2014 to 2023), a larger number of bootstrap samples (i.e., 300) and a reduced search space ratio (i.e., 1/3) was identified for the RSS strategy.

As in the validation set, the net farm income associated with the proposed coverages for the years 2017 to 2014 were compared with the actual realizations using the MSE. As illustrated in Table 2, significant differences were observed in forage yields, prices, and costs during the testing period. Particularly, producers experienced volatile yields and rising hay prices and production costs. In addition to the proposed coverage selection strategies, a Naïve coverage was evaluated as reference. This Naïve coverage consists of allocating 1/3 of the percent of value to the first, third, and fifth available index intervals and selecting a coverage level of 90% and a productivity factor of 150%. Similar Naïve strategies have been evaluated in previous applications (Graff et al., 2021, 2023). Note that the Naïve strategy is a special case of the PS model in which it is possible to assign an equal percent of value to each of the selected index intervals (minimizing the penalization) to obtain the target compensation. However, the Naïve coverage is not entirely comparable with the proposed methods since it targets a fixed return, in this case different from the return sought by the other selection methods. Also, average payouts for Grid 12735 are reviewed to illustrate the actual indemnity payments received by farmers (USDA RMA, 2024a).

4.5 Results

The performance of the four proposed coverage selection models, mean-variance (MV), penalized selection (PS), bagging selection (BS), and random subset selection (RSS), was evaluated in protecting expected forage wheat returns during 2017 to 2024 in Grid 12735 of Texas. The optimal coverages proposed for each model are presented in Table 3. These coverages were intended to obtain the expected net indemnity and net farm income for each year in Table 4 and were derived using actuarial information related to each year. The expected indemnity payments needed to achieve the target returns in Table 2 for the years 2020, 2021, 2022 and 2024 were outside the feasible range. Therefore, for these years, we used the 95^th percentile of the expected payouts as a proxy for expected indemnities. Note that the expected indemnity from the Naïve model is higher than the corresponding 95^th percentile levels, making it a riskier alternative.

In general, the suggested coverages by the MV and PS strategies were more similar to each other and different than the counterpart selections obtained from the BS and RSS models (Table 3). All models chose a 90% coverage level in all years evaluated. Furthermore, and as designed in the Annual Forage program, the productivity factor was the main parameter used to adjust the forage value to be protected. Particularly, the productivity factor increased with the target indemnity. Relatively higher productivity factors were selected between 2020 and 2024, when higher indemnities were sought. In terms of model selections, the RSS chose the highest productivity factors and the BS the lowest.

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Optimal coverage by model, 2017–2024.

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Regarding the allocation of the percent of value, three index intervals were always selected to meet the program restrictions (Table 3). The most frequently chosen index intervals were September–October, December–January, and February–March. On the other hand, the least frequently selected index intervals were November–December and January–February. Interestingly, the first available index interval was always chosen, September–October for 2017 to 2013 and October–November for 2024. Also, the MV and PS models, as well as the BS and RSS methods, always chose the same index intervals. For instance, the BS and RSS strategies selected the September–October, December–January, and February–March index intervals from 2017 to 2023. Therefore, it may be appropriate for the Annual Forage program to re-evaluate the number of index intervals available to simplify the coverage selection process. Among the selected index intervals, except for 2024, a higher and similar percent of value was allocated to the outer index intervals and a relatively small share to the inner index interval. This distribution pattern was consistent across all coverage selection strategies. On average, a relatively higher percent of value was assigned to the index intervals of September–October, January–February, and February–March. In general, for a given year, similar coverages were suggested between related models, i.e., between ensemble learning-based strategies (BS and RSS) and between the PS and MV models. However, more diverse coverages were identified by each model over time as target returns were dynamically updated. Moreover, the similarities between the coverage selections of the MV and PS models suggest that the penalty imposed on the PS strategy had a modest effect on the percent of value allocation. Nonetheless, the distribution of the percent of value of the PS strategy resulted in the allocation with the lowest variance among the selected w_j variables. Overall, the observed distributions of the percent of value highlight the importance of selecting the right index intervals to stabilize farm returns as illustrated in Tables 4 and 5.

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Realized, Expected, and Actual Returns, 2017–2024.

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Out-of-sample mean squared error (MSE) results

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The coverage selections in Table 3 were evaluated using the out-of-sample forage yield and hay price, and actuarial information and rainfall indexes reported for each year. The resulting net indemnities and net farm income from each of the proposed coverages are presented in Table 4. It should be noted that the returns reported in Table 4 are expressed per acre basis and the realized net indemnity payments represent the net benefit associated with each coverage selection model. Overall, net farm income increased by participating in the Annual Forage program. For instance, the net indemnity payments received each year were on average greater than the net farm sales. Therefore, the Annual Forage program could be an effective tool to protect against adverse production and market conditions, as in those years with lower net farm sale values (i.e., 2017, 2018, and 2023). Among all models, the Naïve strategy obtained the highest indemnity payments, but its proposed coverages targeted relatively higher compensations compared to the other models (Table 4). When considering only comparable coverages (those targeting the same return: PS, BS, RSS and MV), the MV strategy yielded the highest indemnity payments. However, the goal of the proposed coverage strategies is not to attain the highest payouts, but to achieve the target return with the minimum risk. By definition, the MV strategy generates the optimal coverage (i.e., the lowest risk for any expected return) on the training set. However, the MV strategy was outperformed by the BS method in the out-of-sample set (Table 5). Even modest differences in performance between models at the acre level can translate into economically significant differences at the farm level. For instance, in Texas, about half of hay and haylage operations manage more than 100 acres (USDA NASS, 2024). Thus, it may be natural to assume that medium and large operations might be more inclined to implement the proposed coverage selection strategies, since the overall benefits they can obtain from the Annual Forage program increase linearly with the number of acres insured.

The MSE was used to compare the out-of-sample performance of the different coverage selection models. Namely, realized net farm income and indemnity payments were compared with their corresponding expected values. The MSE of the net farm income (MSE_Π) can be expressed in terms of the MSEs and the out-of-sample covariance of the net farm sales and net indemnity payments:

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where the circumflex (^) denotes expected values, N_oos is the number of out-of-sample observations, MSE_π and MSE_R are the MSEs of net farm sales and net indemnities, respectively, and COV(·) is the covariance between realized net farm sales and net indemnities received. Note that the first term (MSE_π) in equation 11 is related to the ability to predict future net farm sales, the second term (MSE_R) evaluates the error of the coverage selection strategy used, and the last term measures the degree of correlation between forage production value and compensation of the Annual Forage captured by the chosen coverage. Since MSE_π is independent of the selected coverage, the coverage selection strategy with the lowest MSE_R and highest negative covariance is preferred, since its coverage is the one that most reduces the variability between realized and expected returns. The estimated MSEs and covariance for each coverage selection strategy are presented in Table 5.

Overall, during the test period, the realized net farm income and net indemnity payments were higher than their expected levels (Table 4). When comparing the performance of the coverage selection models, the net indemnities received using the BS strategy were, on average, closer to the expected payouts, resulting in the smallest MSE_R (295.71). The net indemnities of the RSS strategy followed with a MSE_R equal to 345.44. On the other hand, the highest MSE_R values were associated with the Naïve strategy (which aimed for higher payouts) (1322.87), MV (492.87), and PS (446.91) models. Compared with the baseline MV model, the BS strategy reduced the MSE_R by 40.00%. Given the similar percent of value assigned to the selected index intervals across models, the gains in accuracy are primarily attributed to the choice of the index intervals, which was different between the random subspace methods and the PS and MV selection strategies (Table 3).

In the case of net farm income, the smallest MSE_Π was also obtained by the BS strategy (6072.59) and the highest by the Naïve (7875.23) and PS (6174.25) strategies (Table 5). However, the differences between the four proposed coverage selection models in terms of MSE_Π eased. This is attributed to the relatively higher out-of-sample MSE of net farm sales (MSE_Π) (6,582.46) and the higher observed covariance between realized net farm sales and net indemnities from the MV (–470.86) and PS (–427.56) strategies (Table 5). These results highlight two important implications for policyholders and administrators of the Annual Forage program. First, farmers could stabilize their net farm income by effectively participating in the Annual Forage program, but to do so they need to properly formulate their expectations about future forage production and market conditions. The difference between the target return and the expected net farm sales can then be protected by the Annual Forage using appropriate coverage selection strategies, as illustrated in this empirical application. Unfortunately, production data is limited in some locations, which may affect the formation of expectations. Second, the MSE of the net farm income (MSE_Π) can also be reduced by improving the basis risk given by the covariance between net farm sales and indemnity payments received. A reduction in basis risk can be achieved if the rainfall index is highly correlated with forage yield and if the coverage selection strategy used is able to incorporate this relationship. However, a lack of correlation between historical forage yields and the rainfall index has been reported (Maples et al., 2016; ShalekBriski et al., 2020).

The covariance is also an indicator of the model’s ability to serve as an effective risk management decision- making tool by identifying coverages that compensate farmers when net farm sales are low, thereby reducing the variability of returns. In the case of the models evaluated, the MV and Naïve strategies presented the highest and lowest out-of-sample covariance, respectively. The high covariance of the MV model indicates that a systematic selection process can capture the inherent correlation between indemnities and farm sales. Therefore, the low covariance of the Naïve strategy (–15.05) is not surprising given its fixed and independent selection of the coverage decision parameters.

Realized net indemnities resulting from the proposed coverage selection methods were also compared to actual net payments received by farmers from the Annual Forage program over the period 2019 to 2024. For the locality of interest, a coverage level of 85% was chosen by all enrolled acres each year, and no information is available on indemnity payments for 2017 and 2018 (USDA RMA, 2024a). Except for 2019, the proposed coverage selection strategies resulted in higher insurance payouts than the actual average net indemnity payments received by farmers in Brazos County, the county in which Grid 12735 is located (Table 4). The potential for higher and less risky returns could increase farmers’ participation in the Annual Forage program. It is worth mentioning that the region has been experiencing a prolonged drought during the years considered in the study (U.S. Drought Monitor, 2025).