Introduction: Agricultural production is exposed to yield and price risks that can substantially destabilize farm revenue. Conventional crop insurance primarily covers yield losses and may therefore provide limited protection against price fluctuations and joint yield–price shocks. Revenue insurance offers broader protection by incorporating both production and market risks. However, its effectiveness depends critically on accurately modeling the dependence between yields and prices and the dependence among crop revenues within farm portfolios. Conventional approaches to agricultural risk modeling often rely on linear correlation or independence assumptions. Such assumptions may fail to capture nonlinear and asymmetric dependence, particularly in the tails of the distribution and under extreme loss conditions. Copula functions provide a flexible framework for modeling complex dependence structures separately from the marginal distributions of the underlying variables. Moreover, diversification across crops with imperfectly dependent revenues can reduce revenue risk and lower insurance costs. Accordingly, this study develops and evaluates a copula-based multi-crop revenue insurance framework for wheat, barley, and forage maize in Isfahan County, Iran. The study examines yield–price dependence, estimates actuarially based revenue insurance premiums, and evaluates the potential diversification benefits of three two-crop portfolios while accounting for the actual crop-area shares of farmers.
Materials and Methods: The analysis used farm-level data for wheat, barley, and forage maize in Isfahan County during Iranian years 1397–1403, corresponding approximately to 2018–2025 in the Gregorian calendar. The dataset comprised 32,931 observations for wheat, 9,157 observations for barley, and 11,833 observations for forage maize. Three two-crop portfolios were also analyzed: wheat–barley, wheat–forage maize, and barley–forage maize, comprising 3,599, 4,438, and 664 observations, respectively. Crop revenue was calculated as the product of yield and price. Guaranteed revenue was defined as expected revenue multiplied by the selected coverage level. The analysis considered coverage levels of 65%, 70%, 75%, and 80%. Guaranteed prices of 205,000 Iranian rials per kilogram for wheat and 135,000 Iranian rials per kilogram for barley were used. For forage maize, the average observed price of 32,400 Iranian rials per kilogram was applied. Tail-dependence and radial-symmetry tests were first conducted to assess the characteristics of the dependence structures. Marginal distributions were then selected based on goodness-of-fit criteria. The dependence between crop yields and prices was modeled using seven copula families: Gaussian, Student’s t, Clayton, Gumbel, Frank, Joe, and Plackett. The preferred models were selected using the Akaike information criterion, the Bayesian information criterion, and maximum log-likelihood. For each crop and crop portfolio, 100,000 Monte Carlo revenue scenarios were generated. Insurance premiums were estimated using an expected-shortfall approach with a 20% loading factor. The crop-area shares observed in representative farmer portfolios were incorporated into the construction of portfolio revenue distributions. Diversification savings were calculated as the difference between the corresponding single-crop premiums and the multi-crop portfolio premium.
Results and Discussion: The results revealed statistically significant tail dependence between yield and price for all three crops (p < 0.01). The null hypothesis of radial symmetry was rejected, confirming the presence of nonlinear and asymmetric dependence structures. The Gaussian copula provided the best fit for wheat and forage maize, whereas the student’s t copula was selected for barley. Yield–price dependence was negative for all three crops, with the strongest inverse relationship observed for forage maize. At the 70% coverage level, the estimated single-crop premiums were 76.75 million Iranian rials per hectare for wheat, 10.79 million Iranian rials per hectare for barley, and 97.21 million Iranian rials per hectare for forage maize. Thus, forage maize had the highest premium, whereas barley had the lowest. Diversification benefits varied considerably across the crop portfolios. The barley–forage maize portfolio generated the greatest premium reduction, consistent with its weak positive revenue dependence, measured by Kendall’s τ of 0.172, and its Frank-copula dependence structure. At the representative crop-area ratio of 51.4% barley and 48.6% forage maize, the multi-crop portfolio premium at the 70% coverage level was 16.5 million Iranian rials per hectare, compared with 53.6 million Iranian rials per hectare for the corresponding single-crop premiums. This represents an approximate premium saving of 69%. The wheat–forage maize portfolio ranked second in terms of diversification benefits. Its revenue dependence was weak, with Kendall’s τ equal to 0.084, while upper-tail asymmetry was captured by the Joe copula. At the representative crop-area ratio of 56% wheat and 44% forage maize, premium savings reached approximately 62%. In contrast, the wheat–barley portfolio exhibited stronger positive revenue dependence, with Kendall’s τ equal to 0.396, and generated approximately 34% savings at the representative crop-area ratio. Some highly unbalanced portfolios produced negative diversification benefits. This finding indicates that diversification gains depend not only on the number of crops included in a portfolio but also on the dependence structure and the relative area allocated to each crop. Overall, the results demonstrate that multi-crop revenue insurance can improve risk-management efficiency when crop revenues exhibit weak or asymmetric dependence. Conversely, ignoring nonlinear dependence may result in inaccurate premium estimates and inefficient insurance design.
Conclusion: This study demonstrates that copula-based multi-crop revenue insurance can capture complex dependence structures in agricultural risks and contribute to more efficient insurance design. Among the portfolios examined, barley–forage maize and wheat–forage maize generated the greatest premium reductions, whereas wheat–barley provided more limited diversification benefits because of its stronger positive revenue dependence. The findings support the development of crop- and portfolio-specific revenue insurance schemes that reflect regional cropping patterns and actual crop-area shares rather than relying on uniform premium structures. In particular, the Agricultural Insurance Fund could consider piloting multi-crop revenue insurance for barley–forage maize and wheat–forage maize portfolios after validating premium rates and establishing verifiable procedures for measuring crop prices and yields. Before wider implementation, however, operational pilot testing, out-of-sample validation, cost–benefit analysis, and assessment of institutional and technical feasibility are required. Further expansion should also be contingent upon evaluating loss-adjustment procedures, administrative costs, and farmer uptake. Encouraging diversified and complementary crop portfolios may ultimately improve farm-revenue stability and enhance the efficiency of agricultural insurance programs.