A company's financial performance often depends on the uncertain price of a commodity or financial instrument. For example, a lumber distributor might enter into a fixed-price contract for a particular variety of lumber; or a cable manufacturer might have a short position copper; or a firm might have debt whose interest rate is linked to the prime rate. Although modem theories of valuation lead some people to conclude that non-market risk (unsystematic risk) need not be hedged, elimination of all non-essential risk is indeed a desirable goal, so long as it can be achieved at reasonable cost. Companies have good reason to be concerned about the total risk that they face. Total risk causes concern among those whose relationships with the firm are not diversified, such as employees, customers, and suppliers. In addition, reducing operating risks permits a company to accept greater financial risk through leverage, which brings with it the tax advantages of debt. (For a fuller discussion, see Adler and Dumas [1], Shapiro and Titman [14], Doherty [4], and Dufey and Srinivasulu [5].) Quite often, hedging is not just a simple matter of locking in a price or a rate, since the relevant commodity might not be traded a futures market (for example, there are no futures the prime interest rate); even when it is, there may still be differences between the nature of the firm's exposure and the futures contract (the firm may need to deliver June, but there is no June contract). In circumstances like these, a question arises as to whether the futures contracts that are traded can help to minimize the overall risk faced. Note that we do not need to restrict the hedging possibilities to a futures position that exactly matches the unit size of the exposure. For example, it is clear that other things being equal, the more volatile the firm's exposure, the larger its futures position should be. The size of the risk-minimizing position the traded commodity relation to the position that would have been taken had the desired commodity been traded is known as the ratio, and the position is k own as a cross-hedge (see Working [17] and Anerson and Danthine [2]). The appropriate hedge ratio can be determined accurately if the joint probability distribution for all the relevant random variables is known, for then it is simply a problem of mathematics, but the usual practical