Producer Welfare Benefits of Rating Area Yield Crop Insurance

Abstract

Index-based insurance is an innovative concept for evaluating agricultural risks and payouts, which uses an index instead of traditional on-site loss assessment. Area yield insurance, as an index-based approach, is an effective strategy to mitigate moral hazard and adverse selection issues. This study aims to develop area yield insurance as a new insurance plan in Iran for two major crops: wheat and barley. It utilized kernel and joint kernel distributions to price the insurance and assessed producer welfare benefits by comparing the certainty equivalence (CE) of farmers’ utility with and without the policy. Data were collected from East Azerbaijan Province, including county-level yield data for irrigated and rainfed wheat and barley from 1975 to 2019 and 446 individual-level yield data from 2015 to 2019. A two-stage method was used to model yield risk: the first stage fits a trend model, while the second estimates the yield distributions with the detrended data. The results showed a significant difference in premiums calculated by the two distributions, with joint kernel distribution offering the best empirical fit and reasonable premiums. The findings indicate that area yield crop insurance provides positive welfare benefits and should serve as a viable alternative or complement to existing yield insurance plans. The successful implementation of this policy in various countries suggests it can be a suitable risk management program for developing countries like Iran.

1. Introduction

Production in agriculture has always been accompanied by drastic changes in producers’ incomes due to external factors affecting outputs. Farmers always cope with and manage different types of agricultural risks. Crop insurance is one of the most popular risk management tools used by crop producers to deal with these changes or risky situations [1]. There are many crop insurance options that are provided by the Federal Crop Insurance Corporation (FCIC) of the USA for the producers, such as Multiple Peril Crop Insurance (MPCI) that pays indemnities based on the producer’s individual yield or the area yield crop insurance like the Group Risk Plan (GRP) that pays indemnities based on the county-level yields and pays an indemnity when the average county yield falls below a selected trigger level. The area yield crop insurance has some advantages over conventional insurance, which compensates for verified individual yields and losses. In the area yield crop insurance plan, the farmer and insurer have approximately equal information about the distribution of the average county yields, and the classification problems that have led to adverse selection in programs based on individual farm yield are significantly reduced. Moral hazard is eliminated because individual farmers usually have little impact on average area yield [2]. Moreover, better data are typically available from the distribution of area yields than individual farm yields, which will allow insurers to more accurately price the insurance. Since there is no need to manage on-farm loss adjustment, the administration costs (in particular, the cost of assessing losses) will be lower than individual yield crop insurance [1,3,4]. Meanwhile, area yield insurance eliminates the challenges of loss-based insurance, such as verifying individual losses and the high transaction costs of monitoring behavior [5]. All these challenges are more pronounced in developing countries, where information asymmetries, knowledge gaps, and other structural and operational issues are particularly problematic. These issues are even more widespread in such countries. Despite extensive research, there is relatively little evidence to indicate that traditional crop insurance has a positive impact on farmer welfare [6,7]. Like other index-based insurance products, area yield crop insurance is disadvantaged by the basic risk that an indemnity will be only paid when the realized county yields fall below a specified triggered yield, regardless of the farmer’s realized yield. Therefore, the policyholder can experience a loss and yet receive no indemnity. The higher positive correlation between the farm and county yield causes lower basis risk [8,9,10].

Since crop insurance is the cornerstone of domestic agricultural policy in almost-developed and developing countries and vast sums of public monies are used to subsidize crop insurance premiums, there has been a significant amount of literature focused on modeling crop yield distribution, with economists increasingly turning to nonparametric methods due to their advantages. Goodwin and Ker [11], to evaluate the yield risk and insurance premium rates in county-level crop yield, used nonparametric density estimation procedures. They found that nonparametric methods would modify the performance of the crop insurance programs. Chen and Miranda [12] formulated and estimated regime-switching models for Texas county-level dry land cotton yields, and results indicated that they provided a better fit than conventional parametric distribution models, but neither of them generated the premium rate estimation of GRP crop insurance significantly different. Ozaki et al. [13] compared parametric and nonparametric statistical methods for corn, soybean, and wheat aggregate yield data for pricing area yield crop insurance in Brazil. They showed that rates are higher in the nonparametric approach than those in the empirical rate approach. Orlowski [14] simulated area yield insurance indemnities and premium rates in Australia. Park et al. [15] used the Bayesian Kriging for spatial smoothing in crop insurance rating. Liu and Ker [16] investigated the efficacy of using nonparametric Bayesian Model Averaging (BMA) to estimate premium rates for corn, soybean, cotton, and winter wheat by using county yield data in U.S. Kusumaningrum et al. [17] by comparing the cost of insurance and tail risk (VaR and TVaR) in Indonesia, demonstrated that multi-peril crop insurance is not a viable solution due to its high premium and high tail risk, and the area yield insurance is a viable option. Sethanand et al. [18] developed an area yield crop insurance for Jasmine 105 rice at a provincial level and applied the machine learning algorithm to evaluate indemnity payment and premium applicable.

While there are attempts at rate-making area yield insurance plans at the farmer level or county level, evidence is scarce on measuring the welfare benefits of crop insurance. In this context, Adhikari et al. [19] examined the impact of small samples and related policy provisions on the producer welfare benefits of individual-level yield insurance. They indicated that sampling variability in Actual Production History yields could potentially reduce producer welfare. Chung [20] analyzed the social welfare effects of crop revenue insurance for five crops in Korea and found that it led to producer welfare and consumer surplus increases to 5.9 billion tons and 3.0 billion tons, respectively. Adhikari and Luitel [21] conducted a comparative analysis of welfare using two options (yield substitution and yield exclusion) for yield and revenue protection insurance with different risk profiles. They found that the yield exclusion provides a higher welfare gain to producers and is likely to replace yield substitution. Ye et al. [22] evaluated the performance of area yield crop insurance and farm yield crop insurance using farm-level yield data in China, focusing on their effects on farmers’ welfare and their cost-effectiveness in terms of government subsidy.

Furthermore, several farm-level studies indicate that farmers have a high potential demand for index-based insurance plans. Weather and area yield index insurance constitute the two main index-based insurance types currently on the insurance market [23]. Leblois et al. [24] examined how drought frequency affects the demand for index-based insurance in Burkina Faso. Stoeffler and Opuz [25] used experimental and quasi-experimental methods to assess the impact of price, information, and product quality on area yield insurance demand among cotton farmers in Burkina Faso. Shin et al. [26] explored the demand for weather index insurance with basis risk under a prospect theory framework among smallholder farmers in Kenya, finding that prospect theory better explains their insurance demand than expected utility theory. Hossain et al. [27] proposed flood insurance as a climate change adaptation strategy for smallholder farms in Bangladesh, revealing that risk-averse farmers and those aware of high flood risks are more likely to purchase insurance. Additionally, Hossain [28] noted that subsidy policies in Bangladesh are insufficient to promote the adoption of these insurance options.

The background review indicates that the majority of studies on the area yield insurance were concentrated in developed countries, while limited research has been conducted in developing countries, especially the Middle East and South Asia [29]. In this context, few scholars have used nonparametric joint distributions or assessed the producer welfare benefits of insurance. In developing countries, particularly in arid and semi-arid regions, farmers are becoming more vulnerable to climate change and have fewer tools for risk management. The area yield insurance offers an affordable alternative to traditional coverage, speeding up the distribution of vital payouts that can significantly improve farmers’ livelihoods. Insurers also benefit from this insurance, as it allows for better risk management that is cost-effective and flexible and meets farmers’ and market needs. By utilizing real-time data from weather stations and satellites, insurers can effectively transition to index-based practices for agriculture. Consequently, there is a pressing need for innovative insurance programs in these regions. Hence, this study can provide a suitable guide to premium rate-making for policymakers while also emphasizing its welfare effects to promote farmer participation.

In Iran, agricultural insurance has been offered exclusively through the government institution as the Agricultural Insurance Fund since 1984, and it is composed of a wide variety of agricultural insurance products. The fund offers multi-peril crop insurance, greenhouse insurance, and forestry insurance products. Aquaculture is insured through a special program, and livestock insurance is available against accident and mortality from the epidemic disease. Insurance agents do not play an important role in agricultural insurance practice, and there are no particular organizations or programs for servicing agricultural insurance to small and marginal farmers. Both crop and livestock insurance are voluntary and reinsured by the government and rely only on public reinsurance. The government supports agricultural insurance through premium subsidies (69%) that are available for both crop and livestock insurance programs, and agricultural insurance premiums are exempt from sales taxes. The absence of a suitable method for designing crop insurance is pointed out as one of the main problems for the development of an agricultural insurance market [30]. Therefore, the Agricultural Insurance Fund (like other agricultural insurance institutes in the world) is looking for professional advice to train national experts on actuarial principles applied to calculate premium rates. Besides this, despite the widespread area yield insurance plans in the world, this program is not available in Iran, where traditional yield insurance plans are used. This leads to a high loss ratio and asymmetric information problems, such as moral hazard and adverse selection, affecting both the insurer and the insured. In contrast, area yield insurance effectively addresses these challenges, particularly moral hazard, while reducing administrative costs. By basing payouts on area yield assessments rather than individual farm yield, this approach fosters transparency and reduces the opportunities for the farmers to engage in dishonest reporting. Consequently, such a model not only promotes greater trust within the agriculture sector but also encourages more farmers to participate, ultimately leading to a more robust insurance market in Iran. This insurance program can help the Agricultural Insurance Fund mitigate the moral hazard issue and hence provide a cost-effective insurance program. Therefore, the novelty of this paper is that it attempts to develop the area yield crop insurance as a new insurance plan in Iran for two major crops, including wheat and barley. Additionally, it measures the welfare benefits of this insurance for farmers. Despite the extensive research and practice in agriculture, there is still not much clarity as to how much farmers value it over other methods of risk management [31]. Hence, tackling this problem can establish the foundation for farmers’ participation and adoption of this insurance policy. Unlike previous research that employed the univariate kernel distribution approach, this study will adopt the bivariate kernel distribution by combining the farm- and county-level yield data because the main challenge in designing the index is to achieve a strong correlation between crop yields and the chosen index.

Wheat is a crucial agricultural product that plays a vital role in ensuring food security. Barley, the second-highest cereal under cultivation in Iran, is also essential as it feeds livestock. The Agriculture-Jihad Ministry of Iran reported that crop production was 63.2 million tons in 2021, and wheat had the highest production share (18.9%), which is equal to about 11.9 million tons. However, Iran still imported about 7 million tons of wheat. Barley production in the nation was about 2.36 million tons, and 3.3 million tons were also imported. The government has established a guaranteed price policy for these crops to manage their price risk. However, the yield risk remains a significant concern. The Agricultural Insurance Fund has extensively covered wheat and barley as individual production risks for many years. In the crop year 2020–2021, approximately 60% of the insured level of agronomy subsector belonged to wheat (irrigated and rainfed), and the share of barley (irrigated and rainfed) was 15%.

2. Materials and Methods

Assuming that the area yield risk is insurable, the area yield insurance contract, like any other type of agricultural insurance, is determined by the pair [I(y), π], which is the I(y) is indemnity, y the actual county yield, and π the insurance premium. It is assumed that the insurer determines the insurance premium as an actuarially fair one. An actuarially fair premium means that the insurer’s expected loss from policyholders is equal to the premium received, excluding administrative costs. Expected loss refers to the payments in insurance contracts that the insurer expects to pay to policyholders when the contract is signed. Hence, the actuarially fair premium (π) is defined as follows:

$$I\left(y\right) for all \mathrm{y}\in \left[0,\lambda \mu -y \right]$$

(1)

$$\pi =c\left[EI\left(y\right)\right]$$

(2)

where [EI(y)] is expected insured loss, c(.) = 0, $c^{\prime }\left(I\right)\ge 0$ for all $I\ge 0$, λ coverage level, and μ expected yield [10].

The expected insured loss [EI(y)] can be estimated through Equation (3) [10]:

$$EI\left(y\right)=E max\left[\lambda \mu -y,0\right]=\mathit{Pr}ob\left[y<\lambda \mu \right]\left[\lambda \mu -E\left(y\right|y<\lambda \mu \right]$$

(3)

By using the yield distribution function [$f\left(y\right)]$ and simplifying Equation (3), the fair insurance premium rate is as follows [12]:

$$\pi ={\displaystyle \frac{f\left(y\right)\left(\lambda \mu \right)E[\lambda \mu -(y|y<\lambda \mu )]}{\lambda \mu }}$$

(4)

Based on Equation (3), the actuarially fair premium is obtained by multiplying yield risk, Pro$\left[y<\lambda \mu \right]$(probability of loss), in expected loss, [λμ − E(y|y < λμ]. Therefore. any premium rate-making procedure is associated with measuring yield risk. This study utilized a “two-stage method” to model yield risk.

2.1. Two-Stage Method

Agricultural yields have increased over time due to new technologies, productivity, and efficient methods. Thus, the yields in different years are not comparable to each other. For example, the observed yield ten years ago cannot easily be compared with today’s yield. These technical changes cause challenges for accurate modeling of yield distributions in rating crop insurance products [10,12]. Therefore, a common procedure for modeling yield is a “two-stage” method; the first stage fits a trend model to the data, and the second stage uses the detrended data to model the distribution [32]. A variety of methods for detrending yield data have been adapted, such as Autoregressive Integrated Moving Average models [10,33], linear spline [11,34,35], and first- and higher-ordered polynomials [8,12,36,37]. To model yield trends, a first- or second-order deterministic trend model (in t) can be used as follows:

$$y_t={\alpha }_0+{\alpha }_{1}t+{\alpha }_2{t}^2+u_t$$

(5)

where $y_t$ is the realized yield for wheat and rainfed wheat at the county level in years t and t = 1975, 1976, …, 2019. Series with significant slope coefficients (at the 10% level) were detrended. Then, following Deng et al. [4], Chen and Miranda [12], and Ye et al. [22], detrended county-level yield can be computed by normalizing observed yields for the last year 2019 equivalents as follows:

$${y_t}^{det}={\displaystyle \frac{y_t}{{\widehat{y}}_t}}{\widehat{y}}_T$$

(6)

where ${y_t}^{det}$ is the detrended yield in year t, $y_t$ is the realized yield in year t, and ${\widehat{y}}_t$ is the fitted trend yield in year t and T = 2019.

In the second stage, these detrended series are used for modeling yield risk. For modeling yield distributions, parametric and nonparametric approaches are used in the literature. Scholars have considered different parametric distributions, such as beta and normal distribution [38]; beta, normal, and gamma [39]; normal, logistic, Weibull, beta, and lognormal [36]; normal, lognormal, and beta [11]; normal, lognormal, logistic, beta, burr, gamma, and Weibull [40]; and normal, lognormal, gamma, and Weibull [41]. A major advantage of using a parametric approach is the ease of estimation of the distribution of parameters [42], But in the rating of crop insurance products, these common parametric distributions often present problems such as not being able to model bimodality or multimodality, therefore, some researchers prefer the use of nonparametric methods [4,10], which define the shape of the distribution without a given prior specification. Kernel estimator is used as the nonparametric method to estimate the shape of the conditional yield density and pricing of a crop insurance contract, which will be used in this study too.

The nonparametric kernel density is the vertical sum of densities placed over each observed datum. This kernel estimator distributes the probability communicated with each observation, x_i, by using a small density, a kernel centered at each observation. The individual kernel intervals, or windows, are permitted to overlap. The kernel estimator places jumps or bumps at each real observation, and the sum of the densities is used to form the nonparametric curve. The kernel is a function that determines the weights that are specified for each value based on their distance from the center of the window. There are many types of kernel functions (Gaussian, Uniform, Triangle, Triweight, Epanechnikow, Quartic, Cauchy, Double Exponential, Histogram, and Parzen), but researchers have indicated that the estimated density is not sensitive to the choice of the kernel function in moderately sized samples. The width of the kernel is more important than the specific function chosen for the kernel. This width is critical to determining the shape of the density and is called the “bandwidth” parameter [43,44]. The Gaussian kernel density estimator probability density function (PDF) is as follows:

$$f\left(x\right)={\displaystyle \frac{1}{nh}}{\sum }_{i=1}^{n}K\left({\displaystyle \frac{x-x_i}{h}}\right)$$

(7)

where x_i is the ith observation, n is the number of observations, and h is a bandwidth parameter. Also, $K\left({\displaystyle \frac{x-x_i}{h}}\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{(x-x_i)}^2}{2h^2}}}$ is the Gaussian kernel function. Following Goodwin and Goodwin and Ker [11] and Ozaki et al. [13], Silverman’s [45] rule of thumb for the smoothing parameter, $h=0.9\times \min [standard deviation, \mathrm{int}erquartile range/1.34]\times n^{-0.2}$ is used to select the bandwidth parameter. The kernel smoothing method is used to estimate and simulate the county-level yield probability distributions for each crop and county from the detrended county-level yield. However, the simulated data were obtained using the Monte Carlo approach.

2.2. The Joint Kernel Distribution

Considering the relation between individual and area yield can lead to more accurate premium rates. The limited (5 years) farm-level yield data provide only limited information for each representative farm distribution. Thus, to simulate the data, the Monte Carlo method, which was explained in Deng et al. [4,9], is used. Therefore, for each farm, the shocks ε_is should be calculated by each farm yield, ${\tilde{y}}_{is},$ relative to the simple average of the 5 years of individual farms, ${\overline{y}}_i$. The shocks are calculated as follows:

$${\epsilon }_{is}={\displaystyle \frac{{\tilde{y}}_{is}}{{\bar{y}}_i}}$$

(8)

where s indicates the available individual farm-level yield data for the years between 2009 and 2013. Then, the simulated farm-level yields can be calculated by the combination of these shocks and the detrended county yield data as follows:

$$y_i^{simulate}=y_j^{det}\times {\epsilon }_i^{\prime }$$

(9)

where $y_j^{det}$ is a t × 1 column vector of detrended yield for county j, ${\epsilon }_i^{\prime }$ is a 1 × s row vector of shocks for each individual farm i in the county j, $t=1975, 1976,\dots , 2019,$ and $y_i^{simulate}$ is a t × s matrix of simulated yields for each farm i. By determining the z, which is calculated by t × s as a counter variable for the simulated yields, in this study, each farm has a $z=238$ simulated farm-level yields. Due to the limited farm-level yield data, all the simulated farm-level yields in each county were combined in a vector for estimating a yield distribution for a representative farm in that county ($y_f$). Thus, the stacked vector ($y_f$) for each representative farm f has $R=\sum _{i}z$ simulated farm-level yields. Each element of the county-level yield vector ($y_j$) is repeated for the corresponding element of the stacked vector so that the size of both vectors is the same as each other (R). The joint kernel density function of yield for each representative farm f is calculated as follows:

$$\left(y_f.y_j\right)={\displaystyle \frac{1}{R{∆}_f{∆}_j}}\sum _{r=1}^{R}K({\displaystyle \frac{y_f-y_{fr}}{{∆}_f}}.{\displaystyle \frac{y_j-y_{jr}}{{∆}_j}})$$

(10)

where $y_{fr}$ indicates each element of the R elements of the stacked vector ($y_f$). The $y_{jr}$ indicates each element of the $y_j,$ which was introduced before. The ${∆}_f$ and ${∆}_j$ are the bandwidths, and K (.) is a joint kernel function. These processes are explained in detail in Deng et al. [4].

2.3. Premium Rating

The area yield crop insurance contract pays an indemnity if and only if the realized county yield y falls below a critical yield (in practice a guaranteed price established by the government is used to convert units of production per hectare into monetary units per hectare). The critical yield (total liability) is calculated by the $y_c=\lambda \mu$. The $\mu$ is the insurer’s forecast of the area yield, and the coverage level ($\lambda )$ is between 0.65 and 0.9. Following Deng et al. [4], the county yield forecast ($\mu$) is the predicted value from the deterministic trend model described above for t = 2020. If the regression analyses represented no statistically significant time trend in county yield data, the in-sample average yield was used as the county yield forecast. Indemnity is calculated as follows:

$$Indemnity=\mathrm{Max}\left\{0, {\displaystyle \frac{y_c-y}{y_c}}\right\}$$

(11)

where y is the realized yield and $y_c$ is the critical yield. The expectation of the indemnity function is the actuarially fair premium. The fair premium rate, which is calculated by the ratio of the expectation of the indemnity to total liability with a specific probability density function, is calculated [8,10,11,34] as follows:

$$\pi ={\displaystyle \frac{1}{y_c}}{\int }_0^{y_c}\left(y_c-y\right)f\left(y\right)dy$$

(12)

where $\pi$ is the actuarially fair premium rate. For the actual GRP premium rating, a proportional reserve load is applied to the actuarially fair premium. The area under the density to the left of the guaranteed yield presents the probability of loss. The integral under the kernel density, which presents the probability of loss, was numerically estimated by using a trapezoid rule, following Goodwin and Ker [11].

2.4. Producer Welfare

The producer welfare of an insurance contract can be measured by the certainty equivalent (CE) differences with and without insurance [18,46]. The producer welfare increase, resulting from an insurance contract, is because of risk-reducing, which can stabilize producer income. For calculating the rise in producer welfare, the reduced risk should be converted to monetary value. The concept of certainty equivalent, which is employed as a risk management indicator, is used for measuring the economic value of risk. CE, in its simplest definition, explains the highest certain payment a producer would be willing to take to avoid a risky outcome [47]. Certainty equivalent is connected to the risk avoided. Therefore, the utility function, which can reflect the producer’s risk desire, needs to be defined too. Utility increases when income increases, but the increasing rate decreases gradually [19]. For each farm, the certainty equivalent can be calculated as follows:

$${CE}_i={\left[\int U{(\acute{R}}_i\left)h\right(y_i)d{y}_i\right]}^{-1}$$

(13)

where ${CE}_i$ is the certainty equivalent, $U{(\acute{R}}_i)={\displaystyle \frac{({{\acute{R}}_i)}^{1-\gamma }}{1-\gamma }}$ is the constant relative risk aversion (CRRA) utility function with $\gamma =2$ and $h\left(y_i\right)$ is the marginal kernel yield density for representative farm i. A relative risk aversion coefficient ($\gamma )$ of 2 indicates that the farmer would be ready to give up 1% of his or her profit to avoid a gamble that pays 10% of profit with a probability of 0.5 and costs 10% of profit with a probability of 0.5. Myers has estimated that relative risk aversion for a representative U.S. crop farmer lies within the 1–3 range [3]. This is consistent with other studies that use a relative risk aversion coefficient of 2 [4,8,13,20,46]. ${\acute{R}}_i$ is the revenue for the representative farm and can be shown as follows:

$${\acute{R}}_i=price guarantee\times y_i^{net}$$

(14)

where $y_i^{net}$ is the farmer’s net yield when the insurance and can be calculated as follows:

$$y_i^{net}=y_i+I_i-p_i$$

(15)

where $I_i$ and $p_i$ are the indemnity and the insurance premium for representative farm i. $y_i^{net}$ is equal to $y_i$ when there is no insurance purchasing.

The CE_i is estimated at different coverage levels of the area yield insurance plan, and then the producer welfare benefit is measured by comparing the CEi differences with and without the insurance contract.

2.5. Data

This research was conducted in East Azerbaijan Province, the major agricultural producer in the northwestern region of Iran. With 690,000 hectares, it ranks second in wheat acreage (9.24%) and sixth in barley acreage (108,000 hectares, or 6.25%) in the country. Figure 1 shows the map of East Azerbaijan Province and its counties (right-hand panel) in Iran. Ahar, Tabriz, Sarab, Maragheh, Mianeh, and Hashtroud counties account for 48% of the acreage of irrigated wheat, 61% of rainfed wheat, 44% of irrigated barley, and 48% of rainfed barley in East Azerbaijan. These counties are major producers of these crops in the province, and so they were selected as sample counties. In this study, we were required to enter two types of data: county-level yield data and farm-level yield data. Historical county-level yield data were collected for wheat, barley, rainfed wheat, and barley yields for the period of 1975–2019 in the selected counties, including Ahar, Hashtrud, Maragheh, Miyaneh, Sarab, and Tabriz. These data were gathered from the Statistical Yearbook of Agriculture, which is annually published by the Ministry of Agriculture-Jihad of Iran. The farm-level (individual) yield data were not available for all countries. Therefore, the individual yield data were limited to Ahar, Sarab, and Maragheh counties, where the data were available. Since the individual data were limited to these three counties, the analysis of the joint kernel distribution and welfare calculation was limited to these three regions. We collected individual yields for the 5-year duration of yield histories (2015–2019) for wheat, barley, rainfed wheat, and barley. In this case, we collected data from 166 and 157 farmers in Ahar and Sarab counties, as well as from 123 farmers of rainfed wheat and rainfed barley in Maragheh County. These data were obtained from the annual official farm survey plan conducted by the Agriculture-Jihad Organizations in these counties. To ensure the reliability of the data, we cross-referenced the farm-level yield data with the individual yield data obtained through a questionnaire by Afrasyabi et al. [48] and Bayrami [49].

Table 1. The number of wheat, barley, rainfed wheat, and barley farmers.

County/Crop	Wheat	Rainfed Wheat	Barley	Rainfed Barley
Ahar	38	41	48	39
Sarab	53	38	34	32
Maragheh	none *	65	none	58

Source: Agriculture-Jihad Organization, East Azerbaijan Province, Iran. * The farm-level yield data were not available.

provides more information about the number of individual data.

3. Results

Considering the “two-stage” method, first, all the series are detrended, and next, the detrended series were used to estimate the kernel density function for each crop.

Table 2. Descriptive statistics of detrended county-level yields over 1975–2019.

County	Crop	Mean *	Standard Deviation	Minimum	Maximum	Skewness	Kurtosis
Ahar	Wheat	3720	544	2196	4538	−0.85	0.43
	Barley	3363	579	1660	4265	−0.42	1.04
	Rainfed Wheat	1026	232	366	1446	−0.94	1.05
	Rainfed Barley	769	228	300	1310	0.73	0.98
Hashtrud	Wheat	3564	871	2085	4845	0.01	−1.16
	Barley	3027	750	1592	4162	−0.36	−0.80
	Rainfed Wheat	1081	340	359	1707	−0.43	−0.52
	Rainfed Barley	993	373	394	1910	0.14	−0.21
Maragheh	Wheat	3140	734	1554	4461	0.04	−0.48
	Barley	2808	771	376	4806	−0.43	2.41
	Rainfed Wheat	1109	323	202	1753	−0.34	0.42
	Rainfed Barley	1017	217	477	1770	0.57	3.45
Miyaneh	Wheat	3660	909	2190	5612	0.49	−0.32
	Barley	3687	930	1162	6274	0.24	1.27
	Rainfed Wheat	789	258	178	1195	−0.63	−0.53
	Rainfed Barley	951	334	422	2075	0.77	1.82
Sarab	Wheat	3455	681	1563	4639	−0.40	0.04
	Barley	2603	625	1369	3865	0.25	−0.54
	Rainfed Wheat	777	240	117	1309	−0.45	0.59
	Rainfed Barley	684	197	243	1172	−0.26	0.16
Tabriz	Wheat	4138	837	1361	5631	−0.70	2.04
	Barley	2276	590	1320	3451	0.12	−1.38
	Rainfed Wheat	1108	281	113	1688	−1.00	3.58
	Rainfed Barley	815	204	106	1117	−1.22	1.83

Source: authors’ elaborations. * All values are presented in terms of kg/he.

shows descriptive statistics for the detrended yields for all counties and crops. Similar to the findings of Goodwin and Ker [11], Yonar et al. [50] in China and Bangladesh, and Kumar et al. [51] in India, the yields exhibit negative skewness in more than fifty percent of the cases, although positive skewness is reported by researchers such as Chen and Miranda [12]. Negative skewness suggests fatter right-hand-side tails with yields close to the maximum yield observed more frequently than meager yields. Investigating the descriptive statistics of the detrended series in Table 2 indicates that Tabriz County has the highest average yield of wheat, with a maximum of 4138 kg. In contrast, Miyaneh County has the highest average yield of barley, with a maximum of 3687 kg. In addition, Tabriz County had the highest average yield of rainfed wheat, with a maximum of 1108 kg, while Maragheh County had the highest average yield of rainfed barley, with a maximum of 1017 kg. Figure 2 illustrates the estimated kernel distributions for all crops (wheat, barley, rainfed wheat, and rainfed barley), respectively, in Ahar, Hashtrud, Maragheh, Miyaneh, Sarab, and Tabriz counties. The figure indicates that all these series have negative skewness except for Ahar rainfed barley, Hashtrud wheat and rainfed barley, Maragheh wheat and rainfed barley, Miyaneh wheat, barley, and rainfed barley, and Sarab and Tabriz barley, which show a positive skewness. The estimated distributions show significant differences in the area under the density, which shows the probability of loss. The probability of loss for Hashtrud wheat and rainfed barley and rainfed wheat, Maragheh wheat, and Miyaneh rainfed wheat is larger than the others, suggesting higher premium rates for these crops. Maragheh rainfed barley reflects the lowest variance, which suggests that yields close to the mean are more likely than higher and lower values. Some of the densities show evidence of bimodality, such as Hashtrud wheat (with lower yields being more frequent than higher yields), Hashtrud rainfed barley, and Miyaneh rainfed barley (with higher yields being more frequent than lower yields). Tabriz barley shows little evidence of bimodality, in which the yields are close to the mean values.

Table 3. Actuarially and actual premium rates (%) for wheat, barley, rainfed wheat, and barley: univariate kernel distribution.

County	Crop	Coverage	65%	70%	75%	80%	85%	90%
Ahar County	Wheat	Actuarially	1.2	2.0	3.2	4.8	7.0	9.6
		Actual	2.0	3.2	4.6	6.6	9.0	12.2
	Rainfed wheat	Actuarially	2.8	3.5	4.3	5.4	6.5	7.9
		Actual	4.8	5.5	6.3	7.3	8.5	9.8
	Barley	Actuarially	1.0	1.5	2.0	2.8	4.0	5.6
		Actual	1.6	2.2	2.9	3.9	5.2	7.0
	Rainfed barley	Actuarially	2.4	3.2	3.9	5.1	6.6	8.4
		Actual	4.1	5.0	5.9	7.1	8.7	10.4
Hashtrud County	Wheat	Actuarially	2.3	3.5	5.0	6.5	8.5	10.9
		Actual	3.9	5.5	7.3	9.0	11.3	13.4
	Rainfed wheat	Actuarially	6.6	7.1	9.1	10.8	12.3	14.0
		Actual	11.3	11.4	13.5	15.0	16.0	17.3
	Barley	Actuarially	2.5	3.5	4.6	6.0	7.5	9.3
		Actual	4.3	5.5	6.9	8.4	9.8	11.4
	Rainfed barley	Actuarially	7.8	8.5	10.6	12.1	13.6	15.3
		Actual	13.1	13.5	15.8	16.9	17.9	18.8
Maragheh County	Wheat	Actuarially	1.4	2.0	2.8	3.8	5.0	6.7
		Actual	2.4	3.2	4.2	5.3	6.6	8.3
	Rainfed wheat	Actuarially	4.1	5.0	6.0	7.1	8.4	10.0
		Actual	12.0	13.6	14.9	16.1	16.7	17.8
	Barley	Actuarially	3.5	3.8	4.4	5.0	6.0	7.2
		Actual	5.9	6.0	6.5	7.1	7.8	8.9
	Rainfed barley	Actuarially	1.9	2.6	3.7	5.2	7.1	9.7
		Actual	3.2	4.2	5.4	7.1	9.4	12.0
Miyaneh County	Wheat	Actuarially	1.6	2.6	3.9	5.5	7.3	9.5
		Actual	2.8	4.1	5.9	7.6	9.6	11.7
	Rainfed wheat	Actuarially	6.6	7.8	9.2	10.4	11.8	13.3
		Actual	11.3	12.4	13.5	14.4	15.5	16.3
	Barley	Actuarially	1.3	1.6	1.8	2.1	2.6	3.1
		Actual	2.2	2.4	2.7	2.9	3.4	3.8
	Rainfed barley	Actuarially	4.8	6.0	7.3	8.9	10.6	12.3
		Actual	8.2	9.5	10.9	12.3	13.8	15.3
Sarab County	Wheat	Actuarially	1.2	1.7	2.3	3.5	4.9	6.6
		Actual	2.1	2.7	3.5	4.9	6.5	8.2
	Rainfed wheat	Actuarially	5.2	6.3	7.6	8.8	10.1	11.3
		Actual	9.0	10.0	11.2	12.2	13.2	14.4
	Barley	Actuarially	1.6	2.3	3.4	4.8	6.5	8.5
		Actual	2.7	3.7	5.0	6.6	8.4	10.6
	Rainfed barley	Actuarially	5.0	6.1	7.3	8.7	10.2	11.7
		Actual	8.4	9.6	10.7	12.2	13.4	14.5
Tabriz County	Wheat	Actuarially	1.7	2.2	2.7	3.5	4.8	6.3
		Actual	3.1	3.4	4.0	4.9	6.2	7.9
	Rainfed wheat	Actuarially	3.5	4.0	4.8	5.7	6.7	8.1
		Actual	6.1	6.3	7.1	7.9	8.8	10.0
	Barley	Actuarially	1.7	2.7	4.1	6.0	7.9	10.2
		Actual	2.8	4.3	6.2	8.2	10.5	12.6
	Rainfed barley	Actuarially	3.9	4.8	5.9	7.1	8.3	9.9
		Actual	6.7	7.6	8.7	9.8	10.9	12.2

contains the actuarially and actual premium rates for all crops in the counties. These premium rates are reported as a percent of liability. We calculated these premium rates in different coverage levels (65%, 70%, 75%, 80%, 85%, and 90%). Since the area yield crop insurance is not exposed to moral hazard problems, there is no conceptual rationale for imposing constraints on the choice of coverage and scale; hence, it may be easier for farmers and insurers to understand when the scale is to be considered as 100%. As we can see in Table 3, for wheat and rainfed barley, the highest estimated premium rates are in Hashtrud County. For barley, the highest premium rates are estimated for Maragheh, and rainfed wheat belongs to Miyaneh County. These findings align with those of Ye et al. [52], which determined that the premium rates for area yield insurance for wheat range from 2.6% to 5.4% across eight Chinese counties, and the premium rates of rice in Vietnam and India are 5 and 5.1%, respectively [53].

As previously mentioned, in area yield insurance, a higher correlation between county and individual yields leads to more accurate estimates of loss probabilities, premium rates, and indemnity payment schemes. Therefore, assessing and quantifying the relationship between individual and county yields using a bivariate kernel distribution method allows for precise estimations of the yield distribution functions and better Monte Carlo simulations. To achieve this, we estimated the joint kernel distribution. This method allowed us to determine the premium rates for different crops. However, since the individual data were limited to Ahar, Sarab, and Maragheh counties, the analysis of the joint kernel distribution and welfare calculation was limited to these three regions. This study estimated and compared the cumulative distribution functions (CDFs) for univariate and bivariate kernel distributions with the empirical distribution function. The results showed that the bivariate kernel distribution functions align more closely with empirical data, making them preferable for measuring yield risk and pricing area yield insurance. The CDFs, PDF, and empirical distribution functions for rainfed wheat yield, for example, are displayed in Figure 3 and Figure 4 (the CDFs, PDF, and empirical distribution functions for other crops are available upon request.). These figures illustrate that the bivariate kernel distribution function for rainfed wheat yields is preferred over the univariate distribution function as it precisely matches the empirical cumulative distribution function. The resulting calculated premium rates for each crop are reported in

Table 4. Actuarially and actual premium rates (%) for wheat, barley, rainfed wheat barley: bivariate kernel distribution.

County	Crop	Coverage	65%	70%	75%	80%	85%	90%
Ahar County	Wheat	Actuarially	1.0	3.1	3.2	5.6	8.3	9.0
		Actual	1.7	5.0	4.8	7.8	10.7	11.1
	Rainfed wheat	Actuarially	2.7	3.4	4.1	5.1	6.2	7.4
		Actual	4.6	5.4	6.1	7.1	8.1	9.3
	Barley	Actuarially	1.0	1.3	1.8	2.6	3.7	5.4
		Actual	1.7	2.2	2.7	3.5	4.9	6.6
	Rainfed barley	Actuarially	2.2	2.8	3.7	4.8	5.9	7.7
		Actual	3.7	4.4	5.4	6.6	7.7	9.5
Sarab County	Wheat	Actuarially	0.9	1.1	1.5	2.7	4.1	6.0
		Actual	1.6	1.7	2.2	3.7	5.4	7.4
	Rainfed wheat	Actuarially	4.9	6.1	7.2	8.4	9.5	10.9
		Actual	8.4	9.6	10.7	11.6	12.4	13.4
	Barley	Actuarially	1.1	1.5	2.0	3.2	5.2	7.6
		Actual	1.8	2.3	2.8	4.4	7.0	9.4
	Rainfed barley	Actuarially	3.8	5.0	6.1	7.4	8.9	10.5
		Actual	6.5	7.9	9.2	10.4	11.7	12.9
Maragheh County	Rainfed wheat	Actuarially	2.9	3.5	4.2	5.0	6.0	7.6
		Actual	5.0	5.6	6.2	7.0	7.9	9.2
	Rainfed barley	Actuarially	1.7	2.3	3.0	4.3	6.4	9.0
		Actual	3.0	3.6	4.3	6.0	8.3	11.0

. The highest premium rate belongs to wheat in Ahar County, while the highest premiums for other crops are estimated for Sarab County. To determine the statistical significance of the premium rate differences between the two methods, we followed the approach used by Goodwin and Ker [11] and Sherrick et al. [36]. We conducted a paired t-test (

Table 5. Paired t-tests of premium rates of univariate kernel and bivariate kernel distributions.

County	Univariate Kernel Distribution Premium Rates at:	Wheat	Rainfed Wheat	Barley	Rainfed Barley
AharCounty		Bivariate kernel distribution premium rates at coverage (65%)
	Coverage 65%	2.83 ***	0.67 ns	−0.97 ns	1.81 *
		Bivariate kernel distribution premium rates at coverage (70%)
	Coverage 70%	2.36 **	0.78 ns	0.12 ns	1.82 *
		Bivariate kernel distribution premium rates at coverage (75%)
	Coverage 75%	3.77 ***	0.91 ns	1.22 ns	1.87 *
		Bivariate kernel distribution premium rates at coverage (80%)
	Coverage 80%	1.98 *	0.67 ns	1.47 ns	1.79 *
		Bivariate kernel distribution premium rates at coverage (85%)
	Coverage 85%	2.71 ***	1.16 ns	1.82 *	3.01 ***
		Bivariate kernel distribution premium rates at coverage (90%)
	Coverage 90%	3.36 ***	1.65 ns	1.62 ns	3.42 ***
SarabCounty		Bivariate kernel distribution premium rates at coverage (65%)
	Coverage 65%	2.45 **	1.98 **	4.11 ***	5.62 ***
		Bivariate kernel distribution premium rates at coverage (70%)
	Coverage 70%	4.75 ***	1.47 ns	6.15 ***	5.11 ***
		Bivariate kernel distribution premium rates at coverage (75%)
	Coverage 75%	6.04 ***	1.36 ns	9.81 ***	4.98 ***
		Bivariate kernel distribution premium rates at coverage (80%)
	Coverage 80%	5.75 ***	1.46 ns	8.41 ***	4.60 ***
		Bivariate kernel distribution premium rates at coverage (85%)
	Coverage 85%	5.41 ***	1.84 *	6.74 ***	4.33 ***
		Bivariate kernel distribution premium rates at coverage (90%)
	Coverage 90%	3.78 ***	2.43 **	4.86 ***	4.33 ***
MaraghehCounty		Bivariate kernel distribution premium rates at coverage (65%)
	Coverage 65%	-	6.12 ***	-	1.49 ns
		Bivariate kernel distribution premium rates at coverage (70%)
	Coverage 70%	-	6.45 ***	-	2.61 ***
		Bivariate kernel distribution premium rates at coverage (75%)
	Coverage 75%	-	6.84 ***	-	4.64 ***
		Bivariate kernel distribution premium rates at coverage (80%)
	Coverage 80%	-	7.21 ***	-	3.96 ***
		Bivariate kernel distribution premium rates at coverage (85%)
	Coverage 85%	-	7.81 ***	-	4.51 ***
		Bivariate kernel distribution premium rates at coverage (90%)
	Coverage 90%	-	7.84 ***	-	4.47 ***

***, **, and * indicate significant at 1%, 5%, and 10%, respectively. ^ns shows not significant.

) and found that the estimated premium rates differ significantly from the kernel and joint kernel distributions at all coverage levels, except for rainfed wheat in Ahar and some coverage levels for rainfed wheat in Sarab and barley in Ahar. This difference is anticipated as the joint kernel distribution takes advantage of the correlation between individual and county yields, leading to more accurate premiums. This aligns with Goodwin and Ker’s [11] and Sherrick et al. [36]’s findings. Figure 3 and Figure 4 validate the match of the kernel distribution with the empirical distribution.

The Producer Welfare Benefits

To assess the impact of a special insurance product on farmers’ welfare, the difference in the certainty equivalent per hectare of each insured was compared with those who were uninsured. The premiums estimated from the joint kernel distribution were used for these calculations. Figure 5 illustrates the amount of producer welfare benefits with and without insurance at various coverage levels for Ahar, Sarab, and Maragheh counties, respectively. This figure shows that the irrigated wheat and barley have significantly higher certainty equivalent than rainfed wheat and barley. This is to be expected because the production risk of rainfed crops is higher than that of irrigated crops. Furthermore, the certainty equivalent of wheat crops is higher than that of barley. The government’s guaranteed purchase policy for wheat may have contributed to this result, which does not apply to barley. This protection reduces price risk for wheat farmers, leaving barley farmers more exposed to income risk. An increase in coverage level clearly enhances farmer welfare, particularly for irrigated wheat and barley. Ultimately, wheat and rainfed wheat showed higher welfare changes than other crops. For instance, in Ahar County, the welfare changes range from IRR 25,489,417 (USD 1 = IRR 251,800 (in 2020)) per hectare at coverage of 65% to IRR 28,388,412 per hectare at coverage of 90% for irrigated wheat. Similarly, for irrigated barley, the welfare change rose from IRR 20,727,212 to IRR 21,514,895 per hectare; for rainfed wheat, from IRR 11,044,319 to IRR 11,411,298 per hectare; and for rainfed barley, from IRR 6,856,590 to IRR 7,108,321 per hectare. Adhikari et al. [46] showed the certainty equivalent differences for the APH insurance of cotton in Texas and corn in Illinois at the 85% coverage level of USD 46.91 and USD 25.19. In Burkina Faso, Serfilippi et al. [54] found that the certainty equivalent difference for the cotton farmers’ crop insurance is CFA 146,000. Ye et al. [52] indicated that, under the area yield insurance plan, changes in Chinese wheat farmer welfare (without subsidy) are CNY 1.35 and CNY 16.33 at the 50th and 95th percentiles, respectively. In Ghana, Gallenstein and Dougherty [55] demonstrate that area yield index insurance outperforms area revenue index insurance by comparing their CE.

Table 6. The producer welfare changes of the area yield crop insurance.

	Coverage	Wheat	Barley	Rainfed Wheat	Rainfed Barley
Ahar County	65%	0.85	0.52	1.12	0.78
	70%	1.82	0.73	1.54	1.11
	75%	3.08	1.02	2.10	1.54
	80%	5.46	1.68	2.66	2.38
	85%	8.12	2.66	3.50	3.22
	90%	12.32	4.34	4.48	4.48
Sarab County	65%	0.57	0.48	2.38	1.82
	70%	1.82	0.67	3.22	2.66
	75%	3.78	0.95	4.20	3.36
	80%	6.58	1.82	5.32	4.48
	85%	10.36	3.50	6.44	5.74
	90%	15.40	5.46	7.84	7.14
Maragheh County	65%	-	-	1.04	0.88
	70%	-	-	1.40	1.25
	75%	-	-	1.96	1.82
	80%	-	-	2.52	3.08
	85%	-	-	3.36	5.04
	90%	-	-	4.76	7.98

reports the percentage of welfare changes at different coverage levels for wheat, barley, rainfed wheat, and rainfed barley in Ahar, Sarab, and Maragheh counties. Positive welfare changes indicate an increase in welfare resulting from the implementation of the insurance, while negative changes indicate a decrease in welfare. As seen, the welfare changes are positive, indicating that the area yield crop insurance would improve producer welfare. Maximum welfare changes of 12.32% and 15.4% for wheat crops in Ahar and Sarab counties and 5.7% for rainfed barley crops in Maragheh County were observed at a coverage level of 90%. For wheat, barley, and rainfed wheat, the most significant welfare changes of 15.4%, 5.46%, and 7.84%, respectively, were observed in Sarab County at a coverage level of 90%. In this coverage level, the maximum welfare changes for rainfed barley were estimated for Maragheh County at 7.98%. Similarly, Wang et al. [56] used the CE measure and demonstrated that the current crop insurance program in China has increased the farmers’ welfare by a range of 0.2% to 5.1% in five selected provinces.

4. Discussion and Conclusions

This study had two objectives. First of all, it aimed to determine the premium rates for the area yield crop insurance contract and, next, to calculate the welfare benefits of the area yield crop insurance contract for the producers. The premium rates were estimated by nonparametric models through the kernel and joint kernel distributions for East Azarbaijan, Iran. Data on different crops, including wheat, barley, rainfed wheat, and rainfed barley in Ahar, Hashtrud, Maragheh, Miyaneh, Sarab, and Tabriz counties were evaluated. Area yield crop insurance has not been introduced yet in Iran; therefore, this study aimed to evaluate whether this program has any welfare benefits for the producers. In designing area yield crop insurance, it is crucial to consider the correlation between area yields and individual farmer yields. This relationship enhances the accuracy of loss probability assessments, premiums, and indemnity payments. By incorporating individual yields into the estimation of distribution functions and simulations, we can achieve more precise predictions. Findings indicate that estimated premiums for rainfed wheat and barley are higher than those for irrigated crops, reflecting the increased risk associated with rainfed production. The premiums calculated using two different kernel distributions show significant differences. Additionally, the premium rate calculations suggest potential savings compared to current rates in Iran, allowing insured parties to pay lower premiums. This approach also benefits insurers by eliminating the need for on-site farm assessments to calculate losses.

The results showed that the joint kernel distribution produced fair and affordable premium rates. This suggests that farmers might be interested in participating in this new insurance plan, and the government might also support this insurance plan over traditional premium rates of currency yield insurance. Comparing these premium rates to current traditional crop yield insurance shows that area yield insurance is more cost-effective. Basing indemnity on the county yield index rather than individual performance offers lower premiums and reduced farmer costs while also being economical for agricultural insurance companies in terms of execution and administrative costs. Considering the importance of linking individual and county yields to minimize baseline risk and establish reasonable premiums, policymakers should prioritize the use of joint distribution in the insurance rate-making procedures. Due to the shortage of individual yield data, it is recommended that the Agricultural Insurance Fund and/or Ministry of Agriculture make a comprehensive and assessable database of farm-level yields. Since the loss probabilities and premium rates differ in each county, it is essential to price the insurance contracts at the county level rather than provincially. This approach would allow premiums to account for local production risks, which would reduce adverse selection and enhance risk pooling benefits. Ultimately, such a pricing strategy could boost farmers’ demand for agricultural insurance and lower operational costs for governments.

Moreover, implementing the area yield crop insurance, especially for irrigated crops, offers clear welfare benefits for farmers, reinforcing the value of this insurance type. When the coverage level reaches 90%, the welfare benefits significantly increase. Therefore, policymakers should aim for a 90% coverage level for farmers, while the current insurance plan stands at coverage 65%. These results, together with the successful experiences of developed and developing countries such as the USA and India, make it expected that the agricultural insurance policies in developing countries intend to increasingly adopt this innovative insurance plan to enhance existing programs. Additionally, the insurance premium plays a crucial role in the adoption and efficiency of the insurance market, influencing both insurers’ and farmers’ willingness to engage. By comparing the premiums of traditional crop insurance with area yield insurance and assessing their reasonableness, it is anticipated that a market for area yield insurance will emerge, thereby improving the agricultural insurance market. Thus, area yield crop insurance can be introduced as a proper alternative program in Iran and also other countries. However, offering area yield products to farmers is only possible if some conditions are met in terms of data availability, crop supply chain, and market structure. Furthermore, accurately classifying the county into homogeneous areas and having long-term data for these areas will result in more precise premium ratings and help avoid adverse selection issues. Therefore, it is recommended that insurers or policymakers prioritize the identification and classification of homogeneous areas. It is recommended that future studies focus on the classification of homogeneous risky areas and determining the premium rates for those areas.

It is noted that this study had two limitations. First, the farm-level yield data were relatively limited for longer periods, which could introduce bias in measuring the correlation between county-level and farm-level yield. Second, the farm-level yield data were only available in some major counties, which led us to focus on specific counties.