Flexible Weather Index Insurance Design with Penalized Splines

Abstract

In this article, we propose a flexible framework for the design of weather index insurance (WII) based on penalized spline methods. The aim is to find the indemnity function that optimally characterizes the intricate relationship between agricultural production losses and weather variables and thus effectively improves policyholders’ utilities. We use B-spline functions to define the feasible set of the optimization problem and a penalty function to avoid the “overfitting” issue. The proposed design framework is applied to an empirical study in which we use precipitation and vapor pressure deficit (VPD) to construct an index insurance contract for corn producers in Illinois. Numerical evidence shows that the resulting optimal insurance contract effectively enhances policyholder’s utility, even in the absence of the government’s premium subsidy. In addition, the performance of our proposed index insurance is robust to a variety of key factors, and the general payment structure is highly interpretable for marketing purposes. All of these merits indicate its potential to increase efficiency of the agricultural insurance market and thus enhance social welfare.

ACKNOWLEDGMENTS

The authors acknowledge helpful comments and suggestions from Editor Patrick Brockett and two anonymous referees.

Notes

1 Another index-based risk management solution is weather derivatives (see, e.g., Turvey 2001; Brockett, Wang, and Yang 2005; Manfredo and Richards 2009). Compared to WII, hedging with derivatives has several limitations: first, the availability of weather derivatives relies on a functioning and mature derivative market, which can be especially problematic for developing countries; second, effective use of derivatives requires sophisticated agricultural producers who are familiar with the capital market and are more adept to various hedging strategies; third, the reference index underlying weather derivatives is typically broad-based and thus gives rise to significant spatial basis risk.

2 We test the robustness to different choices of sample period in Appendix A.

3 Although it is not the most commonly adopted weather index in the index insurance design literature, max VPD actually has the highest absolute value of correlation with production loss, among all weather variables in our dataset, including precipitation, dew point temperature, max/min temperature, and max/min VPD.

4 That is, for each dimension of data $\eta ,$ we apply the transformation $\eta \prime =\frac{\eta -\mathrm{\text{min}}(\eta )}{\mathrm{\text{max}}(\eta )-\mathrm{\text{min}}(\eta )}.$

5 It is expected that the effectiveness of index insurance will rely highly on these extreme weather events. To confirm its usefulness for also insuring normal weather conditions, we scrutinize the impact of extreme weather data points in Appendix B.

6 Norm-based loss functions such as L1 and L2 losses, on the other hand, are symmetric to positive and negative residuals and therefore are less reasonable for this problem.

Additional information

Funding

KST acknowledges research funding from Nanyang Technological University’s Singapore University Grant and President’s Chair in Actuarial Risk Management. JZ acknowledges research funding support from the Nanyang Technological University Startup Grant (04INS000509C300) and the Singapore Ministry of Education Academic Research Fund Tier 1 Grant (RG55/20).