n extensive body of recent research in futures markets deals with determining A optimal hedge ratios or minimum variance hedge ratios for decision makers seeking to reduce risk on a single commodity. The standard approach involves the construction of a portfolio model of commodity stocks and futures contracts.' However, many firms, governments, and decision makers face multiple risks simultaneously. Therefore, recent literature addresses how such decision makers might use the portfolio model to determine optimal hedges in multiple futures contracts simultaneously (e.g., , Thompson and Bond (1987), Benninga, Eldor, and, Alexander, Musser, and Mason (1986), and).' None of these studies, however, derives hedge ratios within the constraints of a fixed production relationship which relates the products or commodities being hedged.
The purpose of this article is to extend this literature by deriving a model to generate simultaneous minimum risk hedge ratios and to apply the model to the soybean complex where inputs and outputs have a fixed relationship with each other. Determining hedge ratios on related inputs and outputs independently of each other ignores the inherent risk reduction due to the positive price correlation among these products. It is shown that simultaneous hedge ratios for multiple and related risks are quite different from those determined when each individual risk is considered separately. 'Among the many examples are Rolf0 (1980), Peck (1975), Ederington (1979), and Hill and Schneeweis (1982).
*Early examples of the theoretical development of optimal hedge ratios in the case of multiple cash goods and multiple futures contracts include Heifner (1972) and Anderson and Danthine (1980). Researchers have recognized for some time that a true model is more complex than the twoasset case.