Value-at-risk and models of dependence in the U.S. federal crop insurance program

The federal crop insurance program covered more than 110 billion dollars in total liability in 2018. The program consists of policies across a wide range of crops, plans, and locations. Weather and other latent variables induce dependence among components of the portfolio. Computing value-at-risk (VaR) is important because the Standard Reinsurance Agreement (SRA) allows for a portion of the risk to be transferred to the federal government. Further, the international reinsurance industry is extensively involved in risk sharing arrangements with U.S. crop insurers. VaR is an important measure of the risk of an insurance portfolio. In this context, VaR is typically expressed in terms of probable maximum loss (PML) or as a return period, whereby a loss of certain magnitude is expected to return within a given period of time. Determining bounds on VaR is complicated by the non-homogeneous nature of crop insurance portfolios. We consider several different scenarios for the marginal distributions of losses and provide sharp bounds on VaR using a rearrangement algorithm. Our results are related to alternative measures of portfolio risks based on multivariate distribution functions and alternative copula specifications.