LLL Ltd wants to sell $500,000$ units of asset X in six months. Assume that the standard deviation of the semi$-$annual changes in the price of asset X is $$10\mathrm{.}5.$ The standard deviation of the semi$-$annual changes in the futures price for a contract on an asset Y which similar to asset X is $$12.$ The correlation between the semi$-$annual changes in the price of asset X and the semi$-$annual changes in the futures price for the contract on asset Y is $0\mathrm{.}91.$ The size of one futures contract on asset Y is $5,000$ units.

$($a$)$ Should the company enter long or short futures positions in order to hedge its exposure?

$($b$)$ What is the minimum variance hedge ratio in this case? What does this hedge ratio mean?

$($c$)$ How many futures contracts should be traded?

$($d$)$ What is the hedge effectiveness? How can you interpret this number?