OPTIMAL HEDGE RATIO

The following table presents several situations where historical data has been used to determine the standard 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 for hedging. The actual commodities underlying the futures contracts and the commodities being hedged are not important.

$\backslash$table$[[$#$,$Commodity,Asset Size,Contract Size,$\backslash$table$[[$St$.$ Dev.$],[$of Spot$],[$Prices$]],\backslash$table$[[$St$.$ Dev. of$],[$Futures$],[$Prices$]],\backslash$table$[[$Coef$.$ of$],[$Correlation$]]],[1,$Natural Gas,$98,320$mmBTU,$10,000$mmBTU,$0\mathrm{.}054,0\mathrm{.}077,0\mathrm{.}90],[2,$Silver,$35,150oz,5,0000z,0\mathrm{.}078,0\mathrm{.}133,0\mathrm{.}86],[3,$Soybeans,$16,040$ bu$,5,000$ bu$,0\mathrm{.}078,0\mathrm{.}105,1\mathrm{.}00],[4,$Heating Oil,$6,353bbl,1,000bbl,0\mathrm{.}059,0\mathrm{.}105,0\mathrm{.}73],[5,$Corn,$16,805$ bu$,5,000$ bu$,0\mathrm{.}082,0\mathrm{.}085,0\mathrm{.}95],[6,$Copper,$92,875lb,25,000lb,0\mathrm{.}051,0\mathrm{.}146,0\mathrm{.}85],[7,$Gasoline,$309,288$gal,$42,000$gal,$0\mathrm{.}113,0\mathrm{.}100,0\mathrm{.}76],[8,$Crude Oil,$5,235bbl,1,000bbl,0\mathrm{.}098,0\mathrm{.}062,0\mathrm{.}96],[9,$Gold,$272oz,1000z,0\mathrm{.}070,0\mathrm{.}057,0\mathrm{.}77],[10,$Wheat,$32,055$ bu$,5,000$ bu$,0\mathrm{.}141,0\mathrm{.}066,0\mathrm{.}82]]$

For each scenario, determine

$$ the optimal $($variance$-$minimizing$)$ hedge ratio

the corresponding optimal number of futures contracts that have to be used to hedge the specified 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.