Hedging and Joint Production: Theory and Illustrations

THE TRADITIONAL VIEW of hedging in futures markets depicts an agent with a predetermined position in a single cash good as seeking to avoid price risk by taking the equal and opposite position in a futures contract involving that same cash good (see eg. Hieronymus (1971)). This routine hedge has long been recognized as only a simple form of hedging practice. The literature on futures markets includes several attempts to give a more complete account of hedging (see, Working (1962), Telser (1956), Johnson (1960), and Stein (1961)). Recently, that literature has been extended, in several ways. Anderson and Danthine (1978) and Rolpho (1978) treat the case where the cash good production relations are uncertain. Anderson and Danthine (1979) stress the fact most hedging decisions are akin to what are called cross hedges in the trade; that is, they involve a cash good that differs in type, grade, location, or delivery date from that specified in the futures contract. Consequently, the hedger faces the risk that the difference between the cash and futures price (the basis) will vary. We argued that fact of basis risk means that hedges involving portfolios of futures may be preferable to those involving only a single futures. To that end we presented the theory of optimal hedging with one cash good and many futures markets. The purpose of the present paper is two-fold. First, we generalize our previous treatment of cross hedging to the case of multiple cash goods and multiple futures. The primary reason for this is that in certain markets the hedging participants are likely to hold an array of cash goods. This is the case notably for the interest rate futures where many participants hold portfolios of debt instruments. Second, we illustrate the theory through examples. The first is the case of the storage of agricultural goods. It is emphasized that price uncertainty implies the output decision should be made jointly even if the technical production relations are separable. The second case depicts optimal use of Treasury bond and bill futures for an agent holding a portfolio of debt. In contrast with the storage illustration in which the coefficients of the optimal hedge are found by direct statistical analysis of the cash and futures prices, the optimal positions in interest rate futures are found indirectly by using the theory of the term structure of interest rates and empirical estimates of the yield curve.