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  • View in gallery

    Correlation of geographic covariates.

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    Average interpolated seasonal sums of CDDs (°C) by method.

  • View in gallery

    Distribution of corn yields in Illinois and Iowa by county (bushels acre−1).

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    Payout correlations between USDA Risk Management Agency revenue protection insurance and simulated heat index insurance: 1989–2019.

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    Relative changes in risk premiums as a function of distance to nearest station, and by county.

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    Relative changes in risk premiums for varying degrees of risk aversion: 1–5.

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    Relative changes in quantile risk premiums by revenue quartile.

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    Locations of farms Close and Remote with weather stations.

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Accounting for Geographic Basis Risk in Heat Index Insurance: How Spatial Interpolation Can Reduce the Cost of Risk

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  • 1 Swedish University of Agricultural Sciences, Uppsala, Sweden
  • | 2 Business Economics Group, Wageningen University and Research, Wageningen, Netherlands
  • | 3 Agricultural Economics and Policy Group, ETH Zürich, Zurich, Switzerland
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Abstract

Extreme heat events cause periodic damage to crop yields and may pose a threat to the income of farmers. Weather index insurance provides payouts to farmers in the case of measurable weather extremes to keep production going. However, its viability depends crucially on the accuracy of local weather indices to predict yield damages from adverse weather conditions. So far, extreme heat indices are poorly represented in weather index insurance. In this study, we construct indices of extreme heat using observations at the nearest weather station and estimates for each county using three interpolation techniques: inverse-distance weighting, ordinary kriging, and regression kriging. Applying these indices to insurance against heat damage to corn in Illinois and Iowa, we show that heat index insurance reduces relative risk premiums by 27%–29% and that interpolated indices outperform the nearest-neighbor index by around 2%–3% in terms of relative risk reduction. Further, we find that the advantage of interpolation over a nearest-neighbor index in terms of relative risk reduction increases as the sample of weather stations is reduced. These findings suggest that heat index insurance can work even when weather data are spatially sparse, which delivers important implications for insurance practice and policy makers. Further, our public code repository provides a rich toolbox of methods to be used for other perils, crops, and regions. Our results are therefore not only replicable but also constitute a cornerstone for projects to come.

Significance Statement

Extreme heat is an important threat to crops and one that could be exacerbated by climate change. Heat index insurance that compensates farmers when temperatures reach destructive levels has been promoted to reduce moral hazard or as an option where traditional insurance is difficult to implement, such as in developing countries where transaction costs can be prohibitive. This study is to our knowledge the first paper to simultaneously design heat indices using interpolation, simulating the amount of risk farmers would avoid from purchasing index insurance, and show how the risk management effectiveness of the contracts depend on the access to weather stations. We show that heat index insurance significantly reduces risk to corn producers in two states in the U.S. Midwest and corroborate recent results that interpolated temperature estimates outperform observations at local weather stations.

Leppert’s current affiliation: Durham University Business School, Durham, United Kingdom.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel Leppert, daniel.j.leppert@durham.ac.uk

Abstract

Extreme heat events cause periodic damage to crop yields and may pose a threat to the income of farmers. Weather index insurance provides payouts to farmers in the case of measurable weather extremes to keep production going. However, its viability depends crucially on the accuracy of local weather indices to predict yield damages from adverse weather conditions. So far, extreme heat indices are poorly represented in weather index insurance. In this study, we construct indices of extreme heat using observations at the nearest weather station and estimates for each county using three interpolation techniques: inverse-distance weighting, ordinary kriging, and regression kriging. Applying these indices to insurance against heat damage to corn in Illinois and Iowa, we show that heat index insurance reduces relative risk premiums by 27%–29% and that interpolated indices outperform the nearest-neighbor index by around 2%–3% in terms of relative risk reduction. Further, we find that the advantage of interpolation over a nearest-neighbor index in terms of relative risk reduction increases as the sample of weather stations is reduced. These findings suggest that heat index insurance can work even when weather data are spatially sparse, which delivers important implications for insurance practice and policy makers. Further, our public code repository provides a rich toolbox of methods to be used for other perils, crops, and regions. Our results are therefore not only replicable but also constitute a cornerstone for projects to come.

Significance Statement

Extreme heat is an important threat to crops and one that could be exacerbated by climate change. Heat index insurance that compensates farmers when temperatures reach destructive levels has been promoted to reduce moral hazard or as an option where traditional insurance is difficult to implement, such as in developing countries where transaction costs can be prohibitive. This study is to our knowledge the first paper to simultaneously design heat indices using interpolation, simulating the amount of risk farmers would avoid from purchasing index insurance, and show how the risk management effectiveness of the contracts depend on the access to weather stations. We show that heat index insurance significantly reduces risk to corn producers in two states in the U.S. Midwest and corroborate recent results that interpolated temperature estimates outperform observations at local weather stations.

Leppert’s current affiliation: Durham University Business School, Durham, United Kingdom.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel Leppert, daniel.j.leppert@durham.ac.uk

1. Introduction

Extreme heat events can cause substantial losses in U.S. corn production, and climate change is expected to further exacerbate heat stress (Schlenker and Roberts 2009). Therefore, efficient risk management is crucial in order to protect farmers’ incomes when extreme weather conditions occur. Traditional indemnity-based crop insurance comes at the cost of adverse selection and moral hazard (Glauber 2013; Goodwin and Smith 2013), which puts an additional burden on insurers by incentivizing farmers to take on more risk in production (Annan and Schlenker 2015). There are also monitoring costs associated with on-farm loss adjustment. Therefore, weather index insurance (WII) may complement indemnity-based products, especially for systemic perils such as heat, because information asymmetries between insurer and the insured are minimized (Belasco et al. 2019; Vroege et al. 2019). In WII, the insurance payout is a function of weather rather than observed losses, which makes any discrepancy between payout and loss, that is, basis risk, the main adoption hurdle (Clarke 2016; Woodard and Garcia 2008; Barnett et al. 2008). Therefore, it is key to use weather data that best capture extreme events at the production location to keep basis risk low and the weather index insurance market functioning (Dalhaus and Finger 2016).1

In this article, we draw from the geostatistics literature and show how interpolation (Cressie 1988; Wu and Li 2013; Roznik et al. 2019) reduces the basis risk of heat index insurance. We extend the literature in three dimensions. First, we show that heat index insurance based on interpolated temperature data reduces the financial exposure to heat events and can thus be used to insure farmers, especially when weather station data are scarce. Second, we focus on extreme events in both the yield and temperature distribution by using quantile regression and cooling degree-days. Third, our supplementary code provides a rich toolbox of methods to not only replicate our results but to be additionally used to inform future U.S. heat insurance policies and future research.

We simulate index insurance contracts to insure extreme heat for corn production in 131 counties in Illinois and Iowa. These were chosen because they are among the most important corn producers in the United States (USDA 2019) and have comparable climates. Corn production is identified in Annan and Schlenker (2015) as sensitive to moral hazard with respect to heat stress under a traditional crop insurance scheme. We focus on extreme downside events and construct a 40-yr index of extreme heat by aggregating the number of cooling degree-days (CDDs) above 29°C (84°F) over the growing season (Schlenker and Roberts 2009). Three common interpolation methods of varying sophistication are used to estimate county-level CDD indices from observed CDDs at 53 weather stations. For comparison, a second set of CDD indices is constructed using observed CDDs from the weather station geographically closest to each county. Following Conradt et al. (2015) and Dalhaus et al. (2018), we estimate the effect of extreme heat on corn yields using quantile regression (QR), which allows us to estimate effect sizes toward the lower end of the yield distribution (Koenker and Bassett 1978; Conradt et al. 2015). Contract performance is evaluated using quantile risk premiums (Kim et al. 2014). Quantile risk premium evaluates the cost of risk at specific quantiles of the revenue distribution. To simulate scarce weather observations, we reproduce the analysis using station samples of varying sizes.

Our results show that all contracts significantly lower farmers’ risk premiums relative to the uninsured case. Second, the relative risk reduction from interpolated CDD indices using kriging compared with the nearest-neighbor index appears to increase as distances between the insured county and the nearest weather station increases. Moreover, interpolated weather data are less likely affected by weather station failure and thus provide technical advantages (see also Dalhaus and Finger 2016). We therefore suggest that constructing weather indices using kriging interpolation is an attractive alternative to improve heat index insurance’s effectiveness in reducing the financial exposure to heat risk.

This research adds to previous work in this area by combining a comparison of designs of weather indices with an economic analysis of WII contracts. Roznik et al. (2019) show that kriging interpolation may successfully be used to estimate weather indices, but they do not evaluate the effectiveness of insurance contracts designed this way. Based on a solid microeconomic and econometric foundation with a special focus on downside risks, our results can be useful for the development of functioning heat index insurances in the United States and elsewhere that complement the currently existing indemnity-based yield and revenue protection schemes. In the long run, weather index solutions that provide payments right after weather extremes were measured can replace the currently used area risk protection insurances, which are based on regional yield statistics that provide payouts with a delay of up to 6 months after a loss event occurs (Schnitkey 2014).

The remainder of the article is structured as follows: In section 2 we give a background on (i) heat index insurance and design methods, (ii) the evaluation of risk using quantile risk premiums, (iii) theory and application of interpolation methods, and (iv) our statistical testing procedure. In section 3 we present the temperature and yield data. We go on to present and discuss the results and limitations before we end with a conclusion section that includes implications for policy and future research.

2. Theory and methods

Traditional indemnity insurance protection against yield losses in agriculture is associated with costs arising from moral hazard and on-farm loss adjustment (Annan and Schlenker 2015). Insurance that is based on an underlying index of temperature or precipitation has been proposed as complementary protection against weather-related losses (Finger et al. 2019; Norton et al. 2015). The holder of a heat index insurance contract receives a payout whenever the heat index exceeds the number of cooling degree-days above which crop losses are expected. This number is called the strike level S. This insurance mechanism implies that payouts that are solely dependent on weather measurements can mismatch farm-level yields. All discrepancies between insurance payout and on-farm losses are denoted as basis risk. Sources of basis risk are manifold but can be classified into three main categories. First, temporal basis risk results from a mismatch between the insured time period, in which weather extremes are measured and insured, and the time period in which the insured crop is most vulnerable to the insured weather peril (Deng et al. 2007). More specifically, if the insurance design does not capture critical crop growth phases, the insurance payout can be biased (Dalhaus and Finger 2016). Second, geographical basis risk describes discrepancies between insured weather and on-farm weather, most likely resulting from remote weather stations that insufficiently capture the weather at the farm’s location. Third, design basis risk captures all remaining sources of model risk in the weather index contract, such as the choice of a statistical model that is not suited to estimate the impact of weather extremes on downside yield events (Conradt et al. 2015). Our contribution aims at reducing the geographical part of basis risk. We assume crop yield Yit of farmer i in year t to be a linear function gi(WIit) of the weather index WIit, that is, in our case cooling degree-days:
Yit=gi(WIit)+εit,
where εit is a bundle of all contributors to yield variations that are uncaptured in the weather index WIit, that is, the basis risk. These include production inputs and pests but also weather-related losses that are not included in gi(WIit). Following Conradt et al. (2015), a QR model is used to estimate the effect size of heat on yields gi(WIit) putting a special focus on downside yield events. QR is more effective in representing the tail dependency than a mean-based estimator such as ordinary least squares (Koenker and Bassett 1978). QR shifts the focus away from the conditional mean to the conditional median or any other quantile of interest of the dependent variable distribution, which is yield in our case. The conditional QR model leads to the following minimization problem:
β^(τ)=argminβ[τ×yixiTβ|yixiTβ|+(1τ)*yi<xiTβ|yixiTβ|].
QR minimizes the sum of absolute residuals, which are asymmetrically weighted. The weighting factor depends on the sign of the residuals: positive residuals receive a weighting factor of τ and negative residuals are weighted by (1 − τ). We select τ = 0.3 for the insurance contracts, focusing on the effect size of cooling degree-days on yields below the third decile of the yield distribution. Sensitivity analysis for τ = {0.2, 0.4} is presented in Table 1.
Table 1.

Relative changes in risk premiums, QR on τ = 0.2, 0.3, and 0.4. In parentheses, we show significance levels from Wilcoxon signed rank tests with continuity correction of the following hypotheses: RCNN<Rnoins, RCI<Rnoins, and RCI<RCNN; confidence levels of p < 0.1, p < 0.05, and p < 0.01 are indicated by one, two, and three asterisks, respectively.

Table 1.

The payout function of the heat index insurance contract follows a European call option style, δit = Tit × max{0, WIitSit}, where an insurance payout δit for farm i in year t occurs if the weather index WIit (the number of cooling degree-days) exceeds strike level Sit. The total payoff is then dependent on the difference between Sit − WIit multiplied by the tick size Tit or the payment per cooling degree-day. This ensures that the size of the payout will be proportional to the damages associated with a given level of heat.

The strike level is set such that unless the weather index falls outside this level, the farmer can expect average yield that year. The strike level is given by Sit = g−1(WIit) (Dalhaus et al. 2018). The tick size is equal to the estimated slope coefficient, that is, the marginal impact of a single CDD. To test the different insurance options, we assume a nonprofit, risk-neutral insurance provider that sets the premium φit equal to expected payout or at a fair rate. We use the bootstrap procedure described in Dalhaus et al. (2018) to derive the farm individual fair premium φit based on the burn-rate method. A farm’s utility in year t is assumed to be a function of terminal wealth, which we define as revenue in t plus payout and minus insurance premiums, if applicable. The risk reducing property of the insurance strongly depends on the farm’s risk preference. To consistently account for downside risk aversion, we use a power utility function, following Dalhaus and Finger (2016):
Uik(Wti)={Wti1α1αifα1ln(Wti)ifα=1,
where k is the underlying index. We test six different levels of α = {0, 1, 2, 3, 4, 5} representing a gradient from risk neutrality to very high risk aversion. The expected utility of farm i is the probability-weighted average of these yearly utility values (Chavas 2004).

a. A quantile-based measure of the cost of risk

We evaluate different options of WII based on different weather data inputs. Therefore, we develop a risk measure that enables us to compare the abilities of different WII contracts to reduce the financial exposure to heat extremes, that is, to compare basis risk between contracts. This is a measure of the effectiveness of the insurance in reducing the cost of risk. Thus, we assume a farmer to produce corn only, which results in the profit function πit = Yitφit + δit. Moreover, we consider farmer i to maximize her expected utility E[U(π)] over profits F(π). Assuming insurances to be specifically targeted at risk-averse decision-makers, we choose U() so that it reflects downside risk-averse preferences, that is, concavity. More specifically, a farmer prefers insurance plan A over B if it covers losses in Yit by payouts δit at a given premium φit so that E[U(πA)] > E[U(πB)]. We are therefore able to compare the risk-reducing properties of the insurance options by contrasting the insured with the uninsured and the insured cases with each other. In this line of argumentation, a way to monetarize the ordinal expected utility information is to think of risk-averse decision-makers being willing to spend a proportion R, that is, the risk premium, of their expected profits E[π] to switch from the stochastic profits F(π) to a sure amount CE, the certainty equivalent. Accordingly, R is a function of risk preferences and the risk in F(π). With a functioning insurance we would expect Rinsured < Runinsured. More specifically and intuitively, insurance reduces the willingness to pay for risk reduction. Besides comparing an insurance’s ability to reduce the financial exposure to heat risk over the whole profit distribution, we follow Kim et al. (2014) and Chavas (2004, 2019) and evaluate the risk-reducing capacity over different parts of the profit distribution. Kim et al. (2014) suggest decomposing the risk premium R into its quantile components ΔR. This enables us to extend the view on average risk reduction to additionally look at an insurance’s performance to deal with downside yield events. Accordingly, the risk premium in quantile k of the profit distribution can be represented as
ΔRk(12)U(mk1)i=1K{[F(bi)F(bi1)]U(mi1)}[F(bk)F(bk1)]mk2(12)U(M1)U(M1)[xSk(mk1M1)2dF(x)](16)U(mk1)i=1K{[F(bi)F(bi1)]U(mi1)}[F(bk)F(bk1)]mk3(16)U(M1)U(M1)[xSk(mk1M1)3dF(x)]
where U′() indicates the first derivative of the utility function, U″() indicates the second derivative, and so on; mkl is the lth moment2 of the k-profit quantile; M1 denotes average profits; and F(bk) and F(bk−1) are the upper and lower probability bounds of the kth quantile, respectively.

Using the quantile risk premium measures, we compare the uninsured case with two different WII contracts, each with a separate underlying CDD indices. The first index WIitNN for county i and year t is constructed from the observed seasonal sum of CDDs at the station closest to the center of county i [nearest neighbor (NN)]. The second index WIitI is constructed using estimated seasonal sums of CDDs from interpolation of the station observations over a map of the study area using three interpolation methods. Performance tests of the two index types were done using the full station sample (n = 53) and five samples of randomly selected stations in progressively smaller numbers. This allowed us to evaluate performance with geographically sparse data.

The nearest-neighbor index WIitNN was constructed as follows: Distances between each station j and each county centroid i were calculated using a trigonometric function of the individual latitude ω and longitude λ variables, where (ωj, λj) are the coordinates of station j and (ωi, λi) are the centroid coordinates of county i. The equation for distance is defined as
Dj,i=rcos1[sin(ωj)sin(ωi)+cos(ωj)cos(ωi)cos(λiλj)],
where r is a constant reflecting the radian of a sphere that we use to normalize to standard units (Norton et al. 2013); the distm function in R returns distances in meters.

For every county, one station was selected for which the distance between county center coordinates and station was shortest. Seasonal sums of CDDs over the 30-yr time series measured at this station were then used to put together the index WIitNN.

b. Interpolation of temperature data

For the interpolated index, we compare three different spatial interpolation techniques. Inverse-distance weighting, ordinary kriging, and regression kriging. Inverse-distance weighting (IDW) is the simplest of the three. It estimates the temperature at an unobserved location as a weighted average of observed temperatures. The weights are defined as Wij=(1/dijα)/k=1nj(1/dkjα), where the index i denotes a weather station, j is the index of a reference point, and nj is the number of stations that relate to reference point j. Each measurement is multiplied by the inverse of distance from the station i to the reference point j with the exponent α, or power, which modifies the penalty of distance on weights. The power is arbitrary for IDW and not derived from the spatial data (Yang et al. 2004)

Kriging interpolation addresses this problem. It estimates temperature at an unobserved location by producing a weighted average from station observations based on variogram models of spatial covariance (Cressie 1988; Hengl 2009). The ordinary kriging method exploits spatial patterns in the sample to “fill in the gaps” in a two-dimensional field. Sampled data are interpreted as the result of a random process Z(s) (Chiles and Delfiner 1999). The unknown value Z(s0) to be estimated is interpreted as a random variable located at s0 and the weighted average of values in neighboring locations Z(si), i = 1, …, n, where n is the observation sample size. Like IDW, observations closer to the interpolated point have higher weights than more remote observations. However, ordinary kriging also accounts for variances between points. A variogram model is fitted from an experimental variogram of the geographic locations as a function of semivariance and distance. The variogram is defined as the variance of the difference between field values at two locations i and j. Given a sample of observations Z(si) = zi for i = 1, …, N at locations s = (x, y) in 2D space with coordinates x and y, the sample variogram is given by
γ(h)=12|N(h)|N(h)(zizj)2,
where N(h) is the set of all pairwise Euclidian distances ij = h and |N(h)| is the number of distinct pairs in N(h).

That the variance increases with increasing lag distance implies that at short distances, the values of the Z(s) are similar, but with greater lag distance they become increasingly dissimilar. A monotonic increasing slope indicates that the process is spatially dependent (Chiles and Delfiner 1999). The distance at which the variogram flattens is the range of the model. The geographic interpretation of the range is the distance where values at two locations are no longer correlated. The range depends on the variable studied. For example, temperature has a greater range than does rainfall, which is more local (Norton et al. 2015). For heat index insurance, the choice of index will impact the range of the variogram. It is important to understand where this range exists for each index and how far from a station an insurer should be comfortable selling insurance.

The maximum distance between any two stations in the sample is divided into shorter distances, or bins, containing station pairs incrementally farther apart. A larger number of relatively small bins means a larger number of points on the variogram because there is a larger number of distances between pairs for which variances can be estimated. This makes it easier to fit a model such as the exponential to the sample variogram because its shape is more visible. However, such a variogram with many bins is less statistically robust because with more bins of smaller distance increments there will be fewer location pairs for each distance. A rule of thumb is 30 pairs per bin (Chiles and Delfiner 1999); however, to allow analysis with smaller samples of stations, we limit the bins to include pairs numbering at least a third of the sample size. This achieved statistically robust results.

A limitation of ordinary kriging is that it does not account for geographic or climatological trends. For example, temperature often displays a latitude trend such that in the Northern Hemisphere, temperatures typically decrease as one moves farther north. The result is that two locations some distance apart latitudinally are more dissimilar on average compared with the same longitudinal distance (Hengl 2009). We use regression kriging to account for potential spatial trends. A linear regression model is set up to estimate variation in temperature. Initial explanatory variables included elevation, latitude, and longitude, which are the most common covariates in temperature modeling (Sekulić et al. 2020). Exploratory analysis revealed correlation among variables, shown in Fig. 1.

Fig. 1.
Fig. 1.

Correlation of geographic covariates.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

Elevation increases in the west and north direction. Monthly sums of CDDs decrease in the north direction and with increasing elevation. A negative correlation between CDDs and longitude disappears when controlling for elevation. These results mirror common findings in temperature prediction for the Northern Hemisphere (Hengl 2009; Wu and Li 2013). To avoid overfitting the model, latitude and longitude were dropped after measuring correlations with elevations of 0.81 and −0.86, respectively. We estimate the following model for each month in the growing season and year in the data:
Tj=β0+β1(ELEVj)+εj,
where j is the measurement location. Residuals from the regressions were saved for interpolation. Variogram modeling was performed using the “fit.variogram” function as part of the gstat package in R, version 3.5.3. The function automatically selects the model out of a set specified by the programmer that best fits the experimental variogram. Following Wu and Li (2013), we select the model out of exponential, spherical, and Gaussian that minimizes squared errors with the monthly experimental variograms.

Using the spatial dependence structures, ordinary and regression kriging is performed using the function “krige” in gstat. We interpolate over a 3179-cell grid of Illinois and Iowa where each cell is 11 100 m by 11 100 m, or approximately 6.9 mi2. This resolution achieves satisfactory computing speed of 1.5 min per run, while higher resolution is unhelpful since we calculate one average CDD per county and time step. Distances between stations and counties are measured from the station coordinates to the county centroid coordinates. Each county is therefore given a single estimated value per month in the growing season. For regression kriging, interpolated residuals were added back into Eq. (7), with an elevation raster of the same spatial resolution. Interpolated mean CDD sums over the time period and the digital elevation model are displayed in Fig. 2.

Fig. 2.
Fig. 2.

Average interpolated seasonal sums of CDDs (°C) by method.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

Our gridded interpolation approach allows researchers and practitioners to match the resolution of their interpolated data to the level of aggregation in available crop data, such as county or farm-level yields. The interpolated indices WIitI are constructed by creating a 40-yr time series of yearly estimated seasonal sums of CDDs for each of the 131 counties for which we have corn yield data. As such WIitI is a function of all observations in the station sample, while the nearest-neighbor index WIiNN is made of raw observations from only one station closest to i.

c. Statistical testing

Payouts, premiums, and revenue for the kriging-based and station-based contracts were calculated for each county using the equations outlined above, expressed in yield units. We calculate risk premiums for each of the 131 counties. The risk premium of each insurance option is also calculated for each quartile of the revenue distribution and for each degree of risk aversion. With respect to quantile risk premiums, we perform statistical tests of three hypotheses:
H1A:Rk[F(πCNN)]<Rk[F(πU)],
H1B:Rk[F(πCmI)]<Rk[F(πU)],and
H1C:Rk[F(πCmI)]<Rk[F(πCNN)],
where πU is profit in the uninsured case, πCNN is profit with the nearest-neighbor index, and πCmI is profit with either of the interpolation methods: IDW, ordinary kriging, or regression kriging. The battery of tests was performed on the expected utility per county for six levels of risk aversion, and on the quantile risk premium vectors, as we are interested in whether our insurance designs lower the cost of risk for the farmers. Taking into account the ordinal nature of expected utility values (Dalhaus and Finger 2016), we use a nonparametric one-sided paired Wilcoxon rank sum test.

3. Data

Daily temperature data were collected from the Daily Global Historical Climatology Network (GHCN-Daily), which includes daily measurements of minimum, maximum, and average temperatures over 24-h cycles measured at ground level at weather stations across the world. It is the most frequently maintained and error-corrected single station network (Menne et al. 2012) and is commonly used for temperature interpolation (Durre et al. 2010; Wu and Li 2013). The dataset includes 40-yr time series from 53 stations across Illinois and Iowa from 1980 to 2019. Temperature is supplied in degrees Celsius. Daily summaries were downloaded from 1 May through 31 September based on estimates for the growing season of corn in Illinois and Iowa (USDA Risk Management Agency 2010). Since we are interested in estimating the impact of extreme heat, we consider only daytime high temperatures over the 24-h cycle. Following Annan and Schlenker (2015), we consider extreme heat to be degree-days over 29°C (or 84°F). CDDs over the growing season were then summarized by station to produce 53 unique 40-yr time series. The years 1988 and 2012 were the hottest growing seasons on average, with means of 344 and 304 degree-days over 29°C, respectively, across the sampled stations. Standard deviations range from 31 to 91 CDDs. Geographically, summertime temperatures are highest in the south, shown in Fig. 2.

Data on corn yields were downloaded from the USDA National Agricultural Statistics Service (NASS) ad hoc Quick Stats online search tool, providing users with free access to annual surveys conducted by NASS. Yield data are supplied on a per county level and includes time series from 1980 to 2019 for 131 counties (USDA 2019). There is also a positive time trend in USDA yield data where yields tend to increase over time likely due to technological and breeding progress. Therefore, the yield time series were detrended using an outlier-robust M estimator from the MASS package in R. This is particularly useful for shorter time series and has been found to be a good compromise between high-breakdown (very robust) estimators and the very efficient OLS estimator (Finger 2013). Summary statistics of the yield data show that their distributions differ between years. Geographically, yields are higher in the northern half of the area, with the most productive counties located in central Illinois, shown in Fig. 3.

Fig. 3.
Fig. 3.

Distribution of corn yields in Illinois and Iowa by county (bushels acre−1).

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

4. Results

The index contract payouts correlate positively with heat and drought damage reported to the federal insurance provider in over 95% of counties in the study. This suggests that the index contracts are largely effective at capturing heat and drought risk. Compared with the nearest-neighbor index, the between-county variance in correlation (shown in Figure 4) with insured damages is 13% lower for the kriging index. We compare reductions in risk premiums per county for each of the WII contracts with the uninsured. Second, we compare the three interpolated indices with the nearest-neighbor index. Relative reductions in the cost of risk for the uninsured to purchase either the nearest-neighbor index or each of the interpolated indices, respectively, were evaluated. Figure 5 shows the relative change in risk premiums from three investment decisions: 1) switching from no insurance to the nearest-neighbor contract, 2) from no insurance to each of the interpolation contracts, and 3) from the nearest-neighbor contract to each of the interpolation contracts. Figure 6 shows the reduction in risk premiums from our interpolated indices relative to the nearest-neighbor index for five degrees of risk aversion. In each case, relative improvements are constant as risk aversion increases, although the associated p values of a Wilcoxon paired rank sum test decrease with higher risk aversion (see also Dalhaus and Finger 2016). Comparing the nearest-neighbor index with interpolated indices using IDW, ordinary kriging (OK), and regression kriging (RK), we simulate smaller improvements of up to 0.3%; however, their robustness with respect to changes in risk aversion persists.

Fig. 4.
Fig. 4.

Payout correlations between USDA Risk Management Agency revenue protection insurance and simulated heat index insurance: 1989–2019.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

Fig. 5.
Fig. 5.

Relative changes in risk premiums as a function of distance to nearest station, and by county.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

Fig. 6.
Fig. 6.

Relative changes in risk premiums for varying degrees of risk aversion: 1–5.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

For the full revenue distribution, all four CDD index contracts developed here outperform the uninsured case by reducing the financial exposure to risk. On average, interpolated heat index insurance reduces 29% and nearest-neighbor index insurance 27% of the yield risk at the county level. Quantile risk premiums allow us to decompose the overall reduction in the risk exposure into different parts of the revenue distribution. We find that each of our contracts lead to lower risk premiums relative to the uninsured case for the full temporal distribution of revenue, as well as for the fourth quartile. Ordinary kriging and regression kriging also reliably outperform the nearest-neighbor index in the first revenue quartile.

For the risk in the second and third revenue quartiles we do not find a significant difference between insurance and no insurance. This suggests that revenues in extreme loss years are shifted into the second quartile of the revenue distribution, that is, are shifted from extremely low observations to slightly below average events (Fig. 7 in appendix B).

Fig. 7.
Fig. 7.

Relative changes in quantile risk premiums by revenue quartile.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

In addition, we repeat the analysis for a reduced sample of stations. However, we extend the analysis to six samples (100%–50%, at 10% increments). New interpolation and nearest-neighbor indices are constructed from these smaller station samples using the same methods outlined above. We find modest but statistically robust improvements in the relative risk reduction from interpolation with increasing distance to the nearest station. As shown in Fig. 6, the estimated effect of distance to nearest station on relative risk reduction derived from the IDW index is 3.3 percentage units per percent increase in distance. Similarly, we find an effect of 3.8% for the ordinary kriging index and 4.3% for the regression kriging index.

Our key finding is that the relative reduction in risk premiums when switching from a station-based to kriging-based contract is greater for counties farther away from a weather station. We propose that this result follows from the fact that geographic basis risk is correlated with distance between the local farm and the site where weather is observed. Further, switching from insurance based on an underlying nearest-neighbor index provides the largest relative risk reduction as distance to nearest station increases, although the effect for IDW and OK indices are also statistically significant. These results provide an indication that regression kriging is relatively more successful at reducing geographic basis risk. We propose that this is a result of regression kriging accounting for geographic trends that previous interpolation methods do not capture, such as differences in elevation and climatic trends.

Figure 5 also shows the relative risk premium changes disaggregated by county. Both contracts result in risk reduction for every county relative to the uninsured case. In some instances, the reduction in the risk premium is over 60%. The relative changes in risk premiums from switching from the nearest-neighbor contract to the kriging index contract are more heterogeneous but overall positive.

5. Discussion and conclusions

In this article, we have shown how interpolation can be applied to spatial weather data and that this can improve the attractiveness of heat index insurance in agriculture. In particular, we find that a heat index constructed using kriging interpolation outperforms a nearest-neighbor index in terms of risk reduction and that the difference is greater when distances between weather stations are large and weather observations are limited. We suggest that this is evidence of the geographical part of basis risk and that regression kriging in particular may be successful at reducing this risk. Thus, our methodological contribution extends earlier work on systemic heat index insurance in the United States in three dimensions (see, e.g., Belasco et al. 2019 for recent developments). First, we consider individually fitted contracts that cover aggregated risks at the county level, which reduces aggregation bias compared to fitting a nationwide model (Marra and Schurle 1994; Finger 2012). Going even a step further, Dalhaus et al. (2018) suggest farm-level tailoring of index insurance contracts if yield data are available. Second, we move away from focusing on weather station data and use spatial kriging to remove large parts of the geographical basis risk (see also Dalhaus and Finger (2016) who find no advantages of using interpolated data for cumulative rainfall index insurance). Third, we use quantile regression and heat days (CDD) rather than average growing days to focus on major impacts of extreme weather events (Conradt et al. 2015). Here, various studies have shown that crop yields respond nonlinearly to temperature exposure, which should be accounted for when designing index insurance (Schlenker and Roberts 2009; Tack et al. 2015, 2017).

Our results therefore relate to Norton et al. (2013) who show that nearest-neighbor index portfolios can fail when weather data (here, temperature) are not spatially correlated or the geography is diverse, such as large differences in elevation. The kriging approach deals effectively with these problems by calculating weights explicitly from the spatial dependence structure (Cressie 1988). Using precipitation indices presents a trade-off between index accuracy and spatial basis risk because the occurrence of rainfall on any particular day has a higher spatial correlation than the amount of rainfall, but the amount of rainfall is a better predictor of the yield (Norton et al. 2015).

We here compare the risk reducing properties of different underlying indices for an insurance scheme. By doing so we provide a solid basis for implementing the proposed procedure in marketable applications. Before an insurance can enter the market, further issues must be considered. First, legal restrictions, which are often country or even state specific, must be taken into account. Second, a heat insurance product should be embedded into a broader portfolio of weather indices that are offered to the farmer to ensure that the insurance is tailored to a single farm’s risk exposure (Bucheli et al. 2020). The U.S. federal crop insurance program provides area yield index insurance to farmers that pays out in case of regional (area) yield falling below a threshold level. The program provides payouts to, for example, corn farmers, until 16 May in the year after harvest, when yield statistics are available. For farmers, who face losses in the harvest year, this delay in the insurance payout can cause substantial liquidity issues. In contrast, heat index trigger payouts just after a weather event was measured, which can take place even before harvest. We thus do not consider the Area Risk Protection Insurance program to be a useful alternative to indemnity-based insurance and focus on weather index insurance in our analysis.

Moreover, as shown by Mahul (1999) farmers’ individual yields are heterogeneously correlated to area yields, which results in differences in the optimal coverage a farmer should purchase. Our herein proposed heat insurance can be extended in a way that not only a representative heat index for each county is used, but also heat at the farm location can be simulated. It thus combines the advantages of a heat index insurance in terms of asymmetric information issues and a farm individually tailored insurance that considers risk exposure at the farm location (Vroege et al. 2019).

We deliver important insights for insurers who may consider implementing our proposed design as a complement to existing protection against heat risk.3 In the context of U.S. federal crop insurance, WII provides policy makers with a tool to reduce moral hazard within the subsidized program. The current insurance system for systemic risks relies largely on an area yield protection program that indemnifies farmers with a large delay when regional yield statistics are published. The herein proposed weather index program can provide immediate payouts just after weather events are measured.

Our research focused on weather heath index for corn yields in Illinois and Iowa. The restrictions by crop specificity as well as to geographical location present a limitation that, however, can be addressed in future research. We use county-level average yields and only discuss the impact of aggregation bias on our results. In fact, farm-level or, as increasingly available in precision agricultural research, plot-level yield distributions might include a considerably larger portion of idiosyncratic risks that cannot be observed at higher aggregation levels (Marra and Schurle 1994; Finger 2012). Particularly, heat extremes, which our insurance is specifically aimed at, occur at a larger spatial scale and are thus assumed to affect a large share of farms in a county. Finger (2013) provides a survey on the literature estimating aggregation bias in crop yield data. We note that the absence of farm-level data is a limitation in our study but our finding that the relative risk reduction increases with distance to weather stations suggests that kriging will capture basis risk also when farm-level yields are available.

For future research, our code repository provided with this article delivers a rich toolbox of methods to be used for other perils, crops and regions. Our results are therefore not only replicable but also constitute a cornerstone for projects to come. Moreover, future research could compare interpolated index contracts with other alternatives using farm-level yield data and quantify the impact of aggregation bias.

Acknowledgments

The authors have no other sources of funding or conflicts of interest to disclose.

Data availability statement

All data used in this article, as well as the code to reproduce our results, are publicly available on the author’s github repository (https://github.com/DanielLeppert/Leppert-et-al-2021-Replication-Code-Data). It also provides a guide on how to replicate the results of this article.

APPENDIX A

Tale of Two Farms

Here we explore the insurance outcomes in 2012 of two hypothetical farms purchasing the insurance contracts designed in this article. We look at 2012 because it was the hottest growing season in the past decade, and we know that farmers were in particular need of heat protection. Consider farm Close (latitude 41.0°, longitude −91.9°), situated in Jefferson County in southeastern Iowa, close to the Illinois border. It is located 18.3 mi (29.5 km) from the nearest weather station in our dataset. Also consider farm Remote (latitude 40.9°, longitude −88.56°), in Livingston County, Illinois. It is located 41.3 mi (66.5 km) from its most proximate weather station. Both farms operate 300 acres (120 ha) of land and are primarily corn producers. The farms are mapped in Fig. A1.

Fig. A1.
Fig. A1.

Locations of farms Close and Remote with weather stations.

Citation: Weather, Climate, and Society 13, 2; 10.1175/WCAS-D-20-0070.1

Sums of CDDs for the nearest-neighbor index over the growing season are 106.9 and 88.3 for farms Close and Remote, respectively. Similarly, interpolated CDDs from IDW, OK, and RK are 107.5, 113, and 112.8 CDDs, respectively, for farm Close, and 114, 113.8, and 120, respectively, for farm Remote. Table A1 shows CDDs, strike levels, payouts, premiums, and terminal revenue for our four index contracts (NN, IDW, OK, and RK) for farm Close and farm Remote. Payouts are derived from the difference between observed CDDs at the nearest station, or interpolated CDDs at the farm location (resolution 11 000 m by 11 000 m) and the strike level, estimated from quantile regression on 40 years of temperature data. Premiums are average payouts across the time period. As shown in Table A1, terminal revenue is positive for all contracts (which is expected in an extremely hot year), but the interpolated indices (particularly RK) are especially attractive to farm Remote, with a terminal revenue of $50.4 per acre (1 acre = 0.4 ha) as compared with $34 with the NN contract, assuming a price of $1 bushel. These results mirror our conclusion that interpolated indices are particularly useful for farms far from the nearest weather station, and as such make for a more equitable index insurance scheme when compared with the nearest-neighbor baseline.

Table A1.

Insurance outcomes (USD per acre) for two hypothetical farms in 2012, with 100% coverage. CDDs measured in degrees Celsius above 29.

Table A1.

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1

In the U.S. federal crop insurance program, area yield index insurances (Area Risk Protection Insurance; https://www.rma.usda.gov/en/Policy-and-Procedure/Insurance-Plans/Area-Risk-Protection-Insurance) are available to farmers. These pay out in the case of regional (area) yield statistics being below a threshold level.

2

We follow Kim et al. (2014) and include moments l ∈ {1, 2, 3}, i.e., mean, variance, and skewness.

3

We assume that, depending on the level of risk diversification (both spatially and across perils), an insurance company offering heat insurance to farmers would choose either to pool all of the heat risk or to transfer a certain share of this risk to a reinsurer. Data from the insured weather station in the nearest-neighbor insurance case or from the kriging model can be used for pricing the reinsurance.

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