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Regional Research Project NC-221 Conference

“Financing Agriculture and Rural America: Issues of Policy, Structure and Technical Change”

McLean, Virginia

October 1-2, 2001

Copyright 2001 by author. All rights reserved. Readers may make verbatim copies of this document for non-commercial purposes by any means, provided that this copyright notice appears on all such copies.

The Accuracy of Producers’ Probability Beliefs:

Evidence and Implications for Insurance Valuation by Bruce J. Sherrick* University of Illinois 1301 W. Gregory Dr.

Urbana, IL 61801 ph: (217) 244-2637 fax: (217) 333-2312 e-mail: sherrick@uiuc.edu August 2001 * Associate Professor, Agricultural and Applied Finance, University of Illinois, Dept. of Agricultural and Consumer Economics. The author thanks Dave Lins, Scott Irwin, Gary Schnitkey, and Matt Diersen for helpful comments on earlier drafts. The usual disclaimers apply.

The Accuracy of Producers’ Probability Beliefs:

Evidence and Implications for Insurance Valuation Abstract The accuracy of producer’s probability beliefs is examined through a survey of large cash-grain farmers in Illinois. It is found that their subjective probability beliefs about important weather variables are systematically miscalibrated to the true distributions.

The nature and extent of the differences between their subjective and true probability measures are shown empirically, and through fitted calibration functions. The economic significance of inaccurate subjective probability beliefs is established in the context of insurance valuation by producers. The results demonstrate that significant errors in producers’ risk assessments and insurance valuation arise simply from the fact that producers possess systematically inaccurate probability beliefs.

Keywords: precipitation insurance valuation, probability beliefs, risk assessment Introduction Significant resources have been devoted to the development and evaluation of agricultural risk-management products, with particular attention paid to crop yield, and crop revenue insurance contracts. Numerous studies have carefully examined the empirical distributions of crop yields and prices, and have developed various insurance valuation models that are equipped to deal with the resulting risk specifications (Day; Gallagher; Goodwin and Ker; Ker and Goodwin; Nelson; Stokes). On the behavioral side, moral hazard and adverse selection issues have also been carefully assessed and incorporated into explanations of the performance of popular insurance products, and into empirical and theoretical studies of crop insurance demand (Coble, Knight, Pope, and Williams; Just, Calvin, and Quiggin; Smith and Goodwin; and Skees and Reed; and many others). While the bulk of the applications in agriculture have understandably targeted the large array of Federal Crop Insurance Corporation (FCIC) products, there has also been a rapidly increasing interest in the use of weather derivatives as mechanisms to manage specific agricultural risks. To date, the weather derivative market has developed much more rapidly in energy applications, and in insurance for outdoor public events, but studies that parallel crop insurance methods to evaluate weather insurance are also beginning to appear in the literature (Martin, Barnett and Coble; Dischel; Sakurai and Reardon; Turvey; Changnon and Changnon). Importantly, the vast majority of the existing crop insurance and risk management literature is underpinned with the assumption that producers accurately understand and rationally respond to the risks they face.

This research explores the important, but frequently unexamined assumption that producers possess accurate probability beliefs when evaluating risky variables that affect their financial well-being. To do so, a survey designed to elicit subjective probability beliefs about important weather variables that influence producers’ well-being was administered to a set of producers. The recovered subjective probability beliefs are then compared to actual weather event distributions in both empirical and fitted form. Then, calibration functions are estimated to provide insight into the extent and nature of the differences between the “ true” probability distribution and individuals’ subjective probability measures. Standard precipitation insurance contracts are evaluated to demonstrate the economic significance of the differences between the producers’ belief sets and the underlying true distributions of interest. Weather variables are focused on due to their ubiquity, relevance to crop farmers, impossibility of influence by producers, and widely available existing information to condition decision makers’ priors.

Further, insurance on weather variables naturally limits adverse selection and moral hazard influences, and thus isolates the impacts of inaccurate priors in a relatively straight-forward fashion.

The remainder of the paper is organized as follows. Results are first presented from a survey that was used to elicit subjective climate expectations from a sample of agricultural producers. The producers’ subjective probability beliefs are first compared directly to “ true” probabilities at several points on the underlying distribution. Then, calibration functions are fit to provide insights into the nature of the differences between the subjective and historic probability measures. Thereafter, the implications of the differences are developed in terms their impacts on insurance valuation. A summary and concluding remarks complete the paper.

Expectations of Climate Variables Survey A survey was conducted to recover complete probabilistic descriptions of producers’ climate expectations. Participants were selected for their: 1) cooperation with the Illinois Farm Business - Farm Management (FBFM) record keeping association, 2) proximity to a single weather reporting station (to mitigate the potential effects of widely differing experiences, all were in a territory covered by a single NOAA weather reporting station), 3) being relatively large cash grain operations, and 4) demonstrated understanding of probability concepts. Personal interviewers elicited producers’ perceptions of the long-run probabilities of rainfall at various levels through a series of questions posed in both the cumulative distribution function (CDF) framework and inverse CDF framework. Numerous questions were recast throughout the survey to locate any changes in perceptions or misperceptions of the intent of questions. For example, if a respondent indicated that the level of rainfall at which the 25th cumulative percentile occurred was 2", the enumerator would later ask for the probability that 2" would be exceeded to insure that the respondent replied in a manner consistent with the earlier answer. A pretest was administered to insure comfort and adequate facility with probabilistic concepts, and internal checks were constructed to insure that the respondents’ probability measures were indeed consistent and representative of their beliefs. The survey included approximately 12 categories of variables that affected the producer’ s financial well-being and took approximately one hour plus pretest time per respondent to administer. A total of fifty-four surveys were administered and processed into useable form.1 Among the specific climate variables of interest included in the survey are April rainfall and July rainfall.2 Higher April precipitation is considered by Illinois crop producers to be a negative event as it tends to delay planting. Conversely, July precipitation is a positive event, as it tends to enhances crop growth and reproduction during a crucial phase of reproduction.

These two variables were chosen because of their particular importance to grain farmers, and because the effects on the respondents are of opposite sign thus generating a natural contrast for study of the accuracy of their probability beliefs.

Weather Variable Representations A distributional representation is needed to summarize information from the historic weather data, and to provide a more complete description of each producer’ s subjective probability beliefs. A distribution that has been used extensively in various forms to model precipitation amounts, as a function for business losses, and by the insurance industry as a candidate for loss distributions is the Burr-12 distribution, also sometimes referred to as a 3parameter Kappa distribution in weather applications (Mielke; Mielke and Johnson; Tadikamalla).

The Burr has zero support, may take on a wide range of skewness and kurtosis, and can be used to fit almost any set of unimodal data (Tadikamalla, 1980). The Burr distribution is highly flexible and contains the Pearson types IV, VI, and bell-shaped curves of type-I, gamma, Weibull, normal, lognormal, exponential, and logistic distributions as special cases (Rodriguez; Tadikamalla).

Because of this flexibility, it is widely accepted in the climate literature as a representation for precipitation levels, and was used to represent the true distribution and each producer’ s underlying subjective distribution.3 The Burr probability density function (PDF) and cumulative distribution

function CDF for rainfall, Y, with parameters,, and, are respectively:

–  –  –

at the East Central Illinois weather reporting station were used to estimate the parameters of the true distributions of April and July rainfall using maximum likelihood estimation. Parameters for each producer’ s subjective probability measures for both April and July rainfall were also estimated under the same parametric assumptions using nonlinear least squares between implied and tabulated response quantiles.

Results Figure 1 depicts the subjective beliefs about precipitation levels for a selected set of respondents with differing types of probability beliefs. As can be seen in the graph, different forms of miscalibration or incongruence between historic and subjective measures exist. For example, farmers #5 and #47 believed the density of April precipitation to be more spread out and have a higher median than the true (these two represent the most common responses relative to April precipitation). Farmers #19 and #25 have subjective probability measures that are generally shifted to a lower level than the true, but with a somewhat longer right hand tails.

Respondent #44 displays overconfidence, and a slightly elevated central tendency.

Relative to July precipitation, respondent #25 has a higher median while the others each have subjective beliefs with medians lower than the true. Respondent #47 displays extremely high pessimism with highly overstated probability of zero or no rainfall. Respondent #44 represents a typical response for July rainfall with a median that is below the true and somewhat understated probabilities at the high range. Respondent #5 has fairly accurate probability beliefs relative to July rainfall. The cumulative distribution functions are provided as well for convenience in interpretation.

The respondents depicted in the graphs are not meant to be representative of the entire sample, but were chosen simply to illustrate the nature of the information retrieved and to help understand the types of differences both among their responses and between their individual beliefs and the historic measures.

–  –  –

Table 1 summarizes the farmer responses across the entire sample for both April and July precipitation. Several quantiles are tabulated under which the farmers’ responses are summarized and compared to the actual precipitation values. For example, for April precipitation at the 25th percentile, the precipitation level corresponding to the true distribution is 2.30 inches. In other words, there is a 75% chance of receiving at least 2.30 inches of precipitation in the month of April in this weather reporting district. Of the farmers surveyed, 63% expected more precipitation at the 25th percentile. The average of all responses at the 25th percentile of the distribution was

2.77 inches. Note that the average of the expected precipitation is greater than the true amount at all percentile levels, although by only a slight amount at the 10-percentile level. Clearly, the subjective probabilities elicited from this group of farmer respondents generally overweighted what they perceive as the negative event of excess April precipitation, with the fraction overstating the rainfall higher at levels generally considered less desirable. If the respondents had no systematic bias in their beliefs, then the percentage overstating the median might reasonably have been expected to be around 50%, but the miscalibration of the sample appears to be systematically toward overstated levels of precipitation. The standard deviation across responses at each quantile is also provided to show the degree of agreement among respondents at each level.

The respondents’ subjective probability beliefs July precipitation follow a different – yet still pessimistic – pattern. In this case, more rainfall is considered to be a good event, and the respondents generally understate the likelihoods of occurrence. As can be seen in table 1, only 22% of the respondents overstated the quantity of rainfall at the 25th percentile of the actual distribution. In fact, at each percentile level, the farmers understated the incidence of precipitation,or equivalently, overstate the probability of what would be viewed as the negative event – lack of precipitation.

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