Please show all formulas with explanation. Just do not copy from other website.

You are holding a $5 million portfolio with a beta of 0.80.

a) The S&P500 futures beta is 1 and the current level of the index is 1,000. The futures contract is valued at $250 times the index value at the expiration of the contract. Assuming you can hedge with fractional shares of the index, what futures position would you take to hedge yourself as completely as possible?

b) Suppose you think the market is going to dive, and you want to have the same size portfolio but with a beta of -0.2. What futures position would you take to change your overall market exposure to a beta of -0.2, rather than hedging yourself? You can use fractional shares of the futures.