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The purpose of this paper is to contribute to the empirical evidence about crop yield distributions that are often used in practical models evaluating crop yield risk and insurance. Additionally, a simulation approach is used to compare the performance of alternative specifications when the underlying form is not known, to identify implications for the choice of parameterization of yield distributions in modeling contexts.

Using a unique high-quality farm-level corn yield data set, commonly used parametric, semi-parametric, and non-parametric distributions are examined against widely used in-sample goodness-of-fit (GOF) measures. Then, a simulation framework is used to assess the out-of-sample characteristics by using known distributions to generate samples that are assessed in an insurance valuation context under alternative specifications of the yield distribution.

Bias and efficiency trade-offs are identified for both in- and out-of-sample contexts, including a simple insurance rating application. Use of GOF measures in small samples can lead to inappropriate selection of candidate distributions that perform poorly in straightforward economic applications. The β distribution consistently overstates rates even when fitted to data generated from a β distribution, while the Weibull consistently understates rates; though small sample features slightly favor Weibull. The TCMN and kernel density estimators are least biased in-sample, but can perform very badly out-of-sample due to overfitting issues. The TCMN performs reasonably well across sample sizes and initial conditions.

Economic applications should consider the consequence of bias vs efficiency in the selection of characterizations of yield risk. Parsimonious specifications often outperform more complex characterizations of yield distributions in small sample settings, and in cases where more demanding uses of extreme-event probabilities are required.

The study helps provide guidance on the selection of distributions used to characterize yield risk and provides an extensive empirical demonstration of yield risk measures across a high-quality set of actual farm experiences. The out-of-sample examination provides evidence of the impact of sample size, underlying variability, and region of the probability measure used on the performance of candidate distributions.

The purpose of this paper is to assess the production risk for winter wheat producers in Beijing, China, particularly in its 13 districts.

A parametric approach is used to model wheat‐yield distribution for samples and the Kolmogorov‐Smirnov test is used to choose the most appropriate yield distribution. Parameters of the special yield distribution are estimated through the maximum likelihood estimation approach.

The Burr distribution is found to be the most appropriate parametric distribution to model winter wheat‐production risks for the districts of Beijing, except in the districts of Fengtai and Shunyi. Findings also show that the Johnson family distribution is the most appropriate model for these two districts (SB for the Fengtai District and SU for the Shunyi District). The wheat‐production loss ratios of the Beijing districts are between 6 and 15 percent, which is considered medium range in most regions. The highest production risks are located in the Western regions of Beijing (Mentougou and Fengtai) while the lowest production risk is located in the Southeastern region of Beijing (Daxing District).

To generate an objective yield trend and an accurate production risk assessment, linear moving average, instead of linear (or quadratic) regression, is used in this paper.

The design and pricing of weather‐based insurance instruments is strongly based on an implicit assumption that the dependence structure between crop yields and weather variables remains unchanged over time. The purpose of this paper is to verify this critical assumption by employing historical time series of weather and farm yields from a semi‐arid region.

The analysis employs two different approaches to measure dependence in multivariate distributions – the regression analysis and copula approach. The estimations are done by employing Bayesian hierarchical model.

The paper reveals statistically significant temporal changes in the joint distribution of weather variables and wheat yields for grain‐producing farms in Kazakhstan over the period from 1961 to 2003.

By questioning its basic assumption the paper draws attention to serious limitations in the current methodology of the weather‐based insurance design.

The empirical results obtained indicate that the relationship between weather and crop yields is not fixed and can change over time. Accordingly, greater effort is required to capture potential temporal changes in the weather‐yield‐relationship and to consider them while developing and rating weather‐based insurance instruments.

The estimation of selected copula and regression models has been done by employing Bayesian hierarchical models.

The purpose of this paper is to develop methods for inferring if crop insurance premiums imply yield distributions that are valid according to standard laws of probability and broadly consistent with observed empirical evidence. The authors also survey current premium-implied distributions both before and after conditioning on the producer’s choice of coverage level.

Under an assumption of actuarial fairness, the authors derive expressions for upper and lower bounds for premium-implied yield cumulative distribution functions (CDFs) at loss thresholds for each coverage level. When observed premiums imply a CDF that exceeds one or is not non-decreasing, the authors conclude that premiums cannot be actuarially fair. The authors additionally specify very weak conditions for premium-implied yield CDFs to be consistent with two possible reasonable parametric distributions.

The authors evaluate premiums for the year 2018 for 19,104 county-crop-type-practice combinations, both before and after conditioning on producer’s choice of coverage level. The authors find problems in roughly one-third of cases. Problems are exhibited for all crops evaluated, and are strongly associated with areas with lower expected yields and higher yield variability. At least 40m acres are currently insured under premium schedules that cannot possibly be consistent with valid probability distributions.

The authors make two primary contributions. First, the premium-implied yield CDF bounds the authors derive requires fewer assumptions than previous similar work, while simultaneously placing more stringent conditions on premiums to be consistent with actuarial fairness. Second, the authors show that current US crop insurance premiums cannot possibly be actuarially fair for many cases, reflecting tens of millions of insured acres, which implies sub-optimal producer risk mitigation and inequitable expenditures for producers and taxpayers.

A common feature of certain kinds of data is a high level of statistical dependence across space and time. This spatial and temporal dependence contains useful information that can be exploited to significantly reduce the uncertainty surrounding local distributions. This chapter develops a methodology for inferring local distributions that incorporates these dependencies. The approach accommodates active learning over space and time, and from aggregate data and distributions to disaggregate individual data and distributions. We combine data sets on Kansas winter wheat yields – annual county-level yields over the period from 1947 through 2000 for all 105 counties in the state of Kansas, and 20,720 individual farm-level sample moments, based on ten years of the reported actual production histories for the winter wheat yields of farmers participating in the United States Department of Agriculture Federal Crop Insurance Corporation Multiple Peril Crop Insurance Program in each of the years 1991–2000. We derive a learning rule that combines statewide, county, and local farm-level data using Bayes’ rule to estimate the moments of individual farm-level crop yield distributions. Information theory and the maximum entropy criterion are used to estimate farm-level crop yield densities from these moments. These posterior densities are found to substantially reduce the bias and volatility of crop insurance premium rates.

The purpose of this paper is to bring out the development of a flexible model for nonstationary crop yield distributions and its applications to decision-making in crop insurance.

The authors design a nonparametric Bayesian approach based on Gaussian process regressions to model crop yields over time. Further flexibility is obtained via Bayesian model averaging that results in mixed Gaussian processes.

Simulation results on crop insurance premium rates show that the proposed method compares favorably with conventional estimators, especially when the underlying distributions are nonstationary.

Unlike conventional two-stage estimation, the proposed method models nonstationary crop yields in a single stage. The authors further adopt a decision theoretic framework in its empirical application and demonstrate that insurance companies can use the proposed method to effectively identify profitable policies under symmetric or asymmetric loss functions.

This study evaluates the impacts on gross revenue distributions of the use of alternative crop insurance products across different coverage levels and across locations with differing yield risks. Results are presented in terms of net costs, values‐at‐risk, and certainty equivalent returns associated with five types of multi‐peril crop insurance across different coverage levels. Findings show that the group policies often result in average payments exceeding their premium costs. Individual revenue products reduce risk in the tails more than group policies, but result in greater reductions in mean revenues. Rankings based on certainty equivalent returns and low frequency VaRs generally favor revenue products. As expected, crop insurance is associated with greater relative risk reduction in locations with greater underlying yield variability.

The purpose of this paper is to estimate crop yield densities considering time trends in the first three moments and spatially varying coefficients.

Yield density parameters are assumed to be spatially correlated, through a Gaussian spatial process. This study spatially smooth multiple parameters using Bayesian Kriging.

Assuming that county yields follow skew normal distributions, the location parameter increased faster in the eastern and northwestern counties of Iowa, while the scale increased faster in southern counties and the shape parameter increased more (implying less left skewness) in southwestern counties. Over time, the mean has increased sharply, while the variance and left skewness increased modestly.

Bayesian Kriging can smooth time-varying yield distributions, handle unbalanced panel data and provide estimates when data are missing. Most past models used a two-stage estimation procedure, while our procedure estimates parameters jointly.

The purpose of this paper is to investigate the effects of crop insurance premiums being determined by small samples of yields that are spatially correlated. If spatial autocorrelation and small sample size are not properly accounted for in premium ratings, the premium rates may inaccurately reflect the risk of a loss.

The paper first examines the spatial autocorrelation among county-level yields of corn and soybeans in the Corn Belt by calculating Moran's I and the effective spatial degrees of freedom. After establishing the existence of spatial autocorrelation, copula models are used to estimate the joint distribution of corn yields and the joint distribution of soybean yields for a group of nine counties in Illinois. Bootstrap samples of the corn and soybean yields are generated to estimate copula models with the purpose of creating sampling distributions.

The estimated bootstrap confidence intervals demonstrate that the copula parameter estimates and the premium rates derived from the parameter estimates can vary greatly. There is also evidence of bias in the parameter estimates.

Although small samples will always be an issue in crop insurance ratings and assumptions must be made for the federal crop insurance program to operate at its current scale, this analysis sheds light on some of the issues caused by using small samples and will hopefully lead to the mitigation of these small sample issues.

At the beginning of the production year producers face a complex risk management decision environment given by risks specific to their operation, multiple crop insurance contracts and hedging opportunities. The purpose of this paper is to provide a producer-level framework for risk management decision making, focusing on the interaction between crop insurance and hedging.

The authors develop a Monte Carlo simulation model that generates a producer’s net income (NI) distribution that incorporates historical producer risk, price-yield correlation via a copula, price risk, and production costs. The authors evaluate the NI distribution through a modified Modern Portfolio Theory (MPT) decision framework. The authors use the modified MPT decision framework to explore tradeoffs between expected NI and farm ruin (defined as 1 or 5 percent expected shortfall) from different crop insurance contracts and pre-harvest hedging options.

Only revenue protection and the highest two levels of coverage level exist on the efficient frontier. The level of hedging on the efficient frontier ranges from 0 to 55 percent of Actual Production History. The authors find that increasing coverage level 5 percent (from 80 to 85 percent) negatively impacts the optimal hedging amount by 26 percentage points (from 35 to 9 percent).

The model provides the precise identification of financial benefits from different risk management strategies by incorporating producer-level historical yield data, using a copula to capture yield-price dependency structure and producer production cost in generating the NI distribution. This model can be applied to any producer’s characteristics and data.