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An Applied Sabbatical

An Applied Sabbatical . Lecture VII . Basics of Crop Insurance. Nelson, Carl H. “The Influence of Distributional Assumptions on the Calculation of Crop Insurance Premia.” North Central Journal of Agricultural Economics 12(1)(Jan 1990): 71–8.

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An Applied Sabbatical

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  1. An Applied Sabbatical Lecture VII

  2. Basics of Crop Insurance • Nelson, Carl H. “The Influence of Distributional Assumptions on the Calculation of Crop Insurance Premia.” North Central Journal of Agricultural Economics 12(1)(Jan 1990): 71–8. • In the past, farmers received assistance during disasters (i.e., draught or floods) through access to concessionary credit. • Increasingly during the last 10 years of the 20th century agricultural policy in the United States shifted toward market-based crop insurance.

  3. This insurance was supposed to be actuarially sound so that producers would make decisions that were consistent with maximizing economic surplus. • Following Nelson’s discussion, the loss of a crop insurance event could be parameterized as

  4. Where C is the level of coverage (i.e., the number of bushels guaranteed under the insurance policy, typically 10, 20, or 40 percent of some expected level of yield). • A is the probability that level of yield. • R is the expected value of the yield given that an insured event has occurred. • L is the insurance indemnity or actuarially fair value of the insurance.

  5. Given these definitions the insurance indemnity becomes • This loss is in yield space, it ignores the price of the output. • Apart from the question of prices a critical part of the puzzle is the distribution function

  6. Estimating Distribution Functions of Crop Yields • A. Moss, Charles B. and J.S. Shonkwiler “Estimating Yield Distributions with a Stochastic Trend and Nonnormal Errors.” American Journal of Agricultural Economics 75(4)(Nov 1993): 1056-62. • From Nelson, differences in the functional form of the distribution function imply different insurance premium for producers.

  7. The goal of the selection of a distribution function is for the distribution function to match the actual distribution function of crop yields. • Differences between the actual distribution function and empirical form used to estimate the premium leads to an economic loss: • If a distribution systematically understates the probability of lower return, farmers could make an arbitrage gain by buying crop insurance. • If a distribution systematically overstates the probability of a lower return, farmers would not buy the insurance (it is not a viable instrument).

  8. The divergence between the relative probabilities is functions of the flexibility of the distributions moments. • Expected value: First moment • Variance: Second central moment

  9. Skewness: Third central moment • Kurtosis: Fourth central moment

  10. Each distribution implies a certain level of flexibility between moments. • For the normal distribution all odd central moments are equal to zero, which implies that the distribution function is symmetric. • In addition, all even moments are a function of the second central moment (i.e., the variance). • Moss and Shonkwiler propose a distribution function that has greater flexibility based on the normal (specifically in the third and fourth moments).

  11. This new distribution is accomplished by parametric transformation to normality. • The distribution is called an inverse hyperbolic sine transformation

  12. The transformed random variable is then hypothesized to be distributed normally with a given mean and variance

  13. Comparing Distribution Functions Out-Of-Sample • Norwood, Bailey, Matthew C. Roberts, and Jayson L. Lusk. “Ranking Crop Yield Models Using Out-of-Sample Likelihood Functions.” American Journal of Agricultural Economics 86(4)(Nov 2004): 1032–43. • The basic concept was to evaluate the goodness of yield distribution using a variant of Kullback and Leibler’s information criteria:

  14. Like most informational indices, this index reaches a minimum of zero if the two distribution functions are identical everywhere. • Otherwise, a positive number reflects the magnitude of the divergence.

  15. The NRL model then suggests that a variety of models can be tested against each other by comparing their out-of-sample measure. This measure is actually constructed by letting the probability of an out-of-sample forecast equal 1/N where N is the number of out-of-sample draws.

  16. Ignoring the constants, the measure of goodness becomes negative. The more negative the number, the less good is the distributional function fit. NRL then constructs a number of out-of-sample measures of I.

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