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.


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.


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
L  AC  AR




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.
 Given these definitions the insurance
indemnity becomes
C
L
  C  y  dF  y 



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
dF  y   f  y  dy
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.


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).
 The divergence between the relative
probabilities is functions of the flexibility of the
distributions moments.

Expected value: First moment
1 

 x f  x  dx


Variance: Second central moment
 
C
2

  x    f  x  dx
2
1


Skewness: Third central moment
 
C
3

  x    f  x  dx
3
1


Kurtosis: Fourth central moment
4C 

  x    f  x  dx
1

 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).


This new distribution is accomplished by
parametric transformation to normality.
The distribution is called an inverse hyperbolic
sine transformation
1

2
2
ln   t   t   1 




et 
 t  zt  


The transformed random variable is then
hypothesized to be distributed normally with a
given mean and variance

et ~ f et  , 2

2

1
1  et       2
2


exp  
e

,


 1



t
2

 2

 2


0.45
0.4
0.35
Probability
0.3
0.25
0.2
0.15
0.1
0.05
0
-15
-10
-5
0
5
10
15
Detrended Corn Yield
20
25
30
35
40
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:
 f X 

I 

f
 ln  f X   g X    0
f
g


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.
 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.
Iˆ 
 l N  ln  l N 
N
i 1

 
ln g X  g
  l N ln  l N  
N
l
i 1
 
l
N
l

ln

N
N
N
ˆI  I~   l
N

 

ln g X  g 
N

 ln  g  X   
N
g
i 1
 ln g X  
N
i 1
g
 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.