Using Bayesian Spatial Smoothing and Extreme Value Theory to Develop Area-Yield Crop Insurance Rating

Rating of insurance premiums depends on the probability of events in the tail of the distribution. Extreme value theory provides a promising way to assess tail risk. We assume that crop yield follows a Generalized Pareto Distribution (GPD), which is a family of extreme value distributions that has advantages for modeling rare events. GPD parameters are fitted using county-level historical winter wheat yield (1970-2014). Spatial smoothing with Kriging parameters is used within a Bayesian hierarchical framework that helps overcome a lack of data due to the rarity of extreme events. We assume that the spatial correlation of crop yield is embedded in the parameters of the GPD. To obtain the posterior distribution, we use Metropolis-Hastings (MH) steps within a Gibbs sampler. Maximum likelihood estimates of the GPD parameters are used for candidate density in the MH step. In the process, MCMC chains are run for 100,000 iterations and burn-in for the first 20,000 observations. We use Deviance Information Criterion (DIC) and out of sample performance to evaluate the quality of the model. From the estimated results, we verify spatial correlation in crop yield, which substantially affects estimates of posterior distributions of GPD parameters. We further simulate spatial random effect based on posterior values of Kriging parameters (range and sill) to visualize and verify the form of spatial correlation. Estimated premiums from an existing method from which current premiums are based, tend to underestimate premium rates compared to our new proposed method.