Data and Methods
We analyze the revenue distributions resulting from the use of price and yield futures and
from participation in government support programs by simulating the revenue functions in each
case. A general description of the approach is as follows. Most of the analysis is done under the
assumption that prices and yields follow a lognormal distribution. A vector V, consisting of cash
prices and yields of corn and soybeans is generated by using a linear transformation of i.i.d.
univariate standard normal variate based on a variance-covariance matrix estimated from central
Illinois county level data for corn and soybeans. Futures prices and yields are then generated,
conditional on the corresponding cash prices and yields. Thus each pair of cash and futures is
assumed to follow a bi-variate lognormal distribution, resulting in another vector FV, consisting
of futures prices and yields. Revenue distributions are then found from the two vectors, V and
FV. Another general scenario is considered in which the distributional assumption of yield
lognormality is changed such that yields follow the beta distribution.
Under lognormality for both yields and prices, we first generate V= (pc, yc, ps, ys), where
pc, yc, ps, ys represent cash prices and yields of corn and soybeans with mean vector μ and a
variance-covariance matrix (; μ and ( are defined in terms of changes in natural logs, implying
lognormality of prices and yields in levels and allowing the use of Choleski decomposition for
generating the vector V with the required variance-covariance matrix. The Choleski
decomposition means that for every positive definite square matrix (e.g., ( ), there exists a unique
lower triangular matrix T such that TT ’=(. If X → N(0,1) and T is the matrix from Choleski
decomposition, then W=TX + μ is distributed as N( μ, ( ). We use a matrix of four i.i.d univariate
standard normal random variates with a sample size of 10,000 draws each to obtain W3.
Exponentiating W produces the desired vector V.
A variance-covariance matrix was estimated using sample data on cash prices and yields
for Champaign county, Illinois, for the period 1972-93. Yield data were obtained from various
issues of the Illinois Agricultural Statistics (Illinois Cooperative Crop Reporting Service) and
price data were obtained from the Illinois Agricultural Marketing Service. The estimation was
done using log changes in cash prices and yields. The estimated variance-covariance matrix and
correlation matrix are reported in Tables 1 and 2.
Futures prices and yields corresponding to cash prices and yields are generated using a
procedure suggested by Hull. The procedure is similar to that used for generating the vector V,
differing only in the sense that, instead of (, only pairwise correlation coefficients ( 'i ) are
required. The pairwise correlation coefficients reflect basis risk. When 'i is one, there is no
basis risk and futures and cash processes are identical. As 'i decreases, basis risk increases.
Using vectors V and FV, revenue realizations can be computed for any given set of
expected prices and yields and policy parameters. Revenue from using just cash markets, ∣ur. is
computed as:
3γi I-Ti Γ'Γ',1 1,∙1 ,1, 1
See Tong for further details on this procedure.