A feed dealer can purchase corn, soybeans, sorghum, and wheat that can be stored and sold to livestock farmers later in the year. Assume that in the past 10 years, the unit profits/losses ($) per bushel on the sale of each commodity are as follows:
|
Year
|
Corn
|
Soybeans
|
Sorghum
|
Wheat
|
|
1
|
-0.05
|
-0.01
|
0.05
|
-0.25
|
|
2
|
0.50
|
0.30
|
0.04
|
1.00
|
|
3
|
0.10
|
0.15
|
0.03
|
0.50
|
|
4
|
-0.25
|
-0.20
|
0.10
|
-0.50
|
|
5
|
0.55
|
0.35
|
0.15
|
-0.20
|
|
6
|
0.50
|
0.20
|
0.20
|
1.10
|
|
7
|
0.10
|
0.12
|
0.05
|
0.80
|
|
8
|
-0.40
|
0.01
|
0.04
|
-0.60
|
|
9
|
0.75
|
0.60
|
0.15
|
1.00
|
|
10
|
0.25
|
0.25
|
0.10
|
-0.50
|
|
Average
|
0.21
|
0.18
|
0.09
|
0.24
|
|
Variance-Covariance Matrix of Expected Unit Profit
|
|
|
Corn
|
Soybeans
|
Sorghum
|
Wheat
|
|
Corn
|
0.139
|
0.067
|
0.012
|
0.159
|
|
Soybeans
|
0.067
|
0.049
|
0.005
|
0.078
|
|
Sorghum
|
0.012
|
0.005
|
0.003
|
0.008
|
|
Wheat
|
0.159
|
0.078
|
0.008
|
0.531
|
Assume the feed dealer can buy and store 500,000 bushels of each of the grains.
a. Formulate the LP problem that maximizes expected profit, where the expected profit is the average profit from the 10 years of observations.
b. Formulate Part a as a quadratic risk programming problem, where the objective function is to minimize the total variance-covariance matrix subject to a minimum expected return constraint. Use parametric programming and start off by setting the minimum expected return RHS value to the profit-maximizing solution found in part a.
c. Trace out the E-V frontier for this problem using parametric programming.