where θ is the coverage level (proportion) assured under the revenue assurance program and Z is
the target gross revenue. Note that the target gross revenue applies to individual crop income as
opposed to farm income.
Although the above approach has advantages of tractability, simulating net revenues
based on jointly normal or lognormal prices and yields may not reflect the true data generating
process well. The assumption of normality in crop yields is of particular concern (e.g., Buccola).
The simulation of independent variables with nonnormal distributions can be done easily
for a range of distributions. However, imposing dependence in the construction of continuous
multivariate distributions with specified marginal distributions of individual variables is
challenging. Johnson and Tenenbein propose a solution to this problem using a weighted linear
combination method for constructing families of bivariate distributions F(x,y) with specified
marginal distributions F1(x) and F2(x), and a level of dependence specified by Spearman’ s
coefficient r. A pair of random variables (X,Y) with marginal distributions F1(x) and F2(x) are
generated as follows.
Let U = U and V = cU ’ + (1-c)V , where U and V ’ are independent and identically
distributed with a common density function g(t), and c is a constant in the interval [0,1]. Johnson
and Tenenbein provide the required values of c as a function of r and the particular specification
of g(t). Let X’ = H1(U) and Y = H2(V), where H1(U) and H2(V) are the distribution functions of
U and V respectively. Now define the following.
X = F1-1(X ) = F1-1(H1(U)),
Y = F2-1(Y’ ) = F2-1(H2(V)), (for a positive value of r)
Y = F2-1(1-Y’ ) = F2-1(1-H2(V)), (for a negative value of r).
Since X’ , Y’ and 1-Y are uniformly distributed over the interval [0,1], Johnson and Tenenbein
note that X and Y will have a joint distribution with marginals F1(x) and F2(x) respectively.
Therefore, for the purpose of simulation, all that is required is the knowledge of the two marginal
distributions.
We apply this procedure for generating revenue distributions by drawing from three
bivariate distributions relating (1) cash yields and cash prices, (2) cash price and futures prices
and (3) cash yields and futures yields. We chose the standard normal distribution for the
underlying density function g(t) which is used in the three random generation procedures.
Interdependence between the bivariate distributions is specified in the simulations using c with
g(t). Levels of c are obtained by solving for it in the following function of r:
r = (6/ π)* arcsin (c/2)* [ c2 + (1-c)2 ] (5)
The simulations were performed using SHAZAM (version 7.0) software program using
10,000 trails. An evaluation of the distribution using the Bestfit program (which describes any
given sample data using about 25 alternative distributions providing ranks for the best fitting
distribution) indicated that corn and soybean yields in Champaign county are best described by