The following table presents several situations where historical data has been used to determine thestandard deviations of spot and futures prices and the coefficients of correlation between the two.The table also lists the sizes of the assets to be hedged and the sizes of the futures contracts used forhedging. The actual commodities underlying the futures contracts and the commodities being hedgedare not important.

For each scenario, determine

? the optimal (variance?minimizing) hedge ratio? the corresponding optimal number of futures contracts that have to be used to hedge thespecified amount of the asset? the actual hedge ratio that would realize if the optimal number of contracts is used.

Round your hedge ratios to four decimal places. Keep in mind that the exchanges do not sell fractions of a contract.

St. Dev. St. Dev. of # Commodity Asset Size Contract Size of Spot Futures Coef. of Prices Prices Correlation T. Natural Gas 98,320 mmBTU 10,000 mmBTU 0.054 0.077 0.90 2 Silver 35/150 0Z 5/000 OZ 0.078 0.133 0.86 3 Soybeans 16,040 bu 5,000 bu 0.078 0.105 1.00 Heating Oil 6,353 bbl 1,000 bbl 0.059 0.105 0.73 5 Corn 16,805 bu 5,000 bu 0.082 0.085 0.95 6 Copper 92,875 lb 25,000 lb 0.051 0.146 0.85 7 Gasoline 309,288 gal 42,000 gal 0.113 0.100 0.76 8 Crude Oil 5/235 bbl 1,000 bbl 0.098 0.062 0.96 9 Gold 272 0Z 100 0Z 0.070 0.057 0.77 10 Wheat 32,055 bu 5,000 bu 0.141 0.066 0.82