$3.$ Malin Investment Trust manages a well$-$diversified $$500$ million equity fund. In March $2024,$ the Trust was concerned about a market decline, perhaps lasting until June. The March value of the portfolio the Trust manages was $$500$ million and the portfolio had a beta of $1\mathrm{.}25.$ Suppose Malin Investment Trust decided in March to maintain its portfolio beta at $1\mathrm{.}25$ if the market were greater than $2,500$ in June, but have a beta of $0\mathrm{.}5$ if the market were less than $2,500.$ In March, the S&P $500$ spot index was trading at $2,500$ and a June S&P $500$ spot call and put contracts with exercise prices of $2,500$ and $$100$ multiplier were trading at $50.$

a$.$ Using the price sensitivity model, determine how many S&P $500$ call or put contracts the Malin Investment Company would have needed in March in order to decrease her portfolio beta to $0\mathrm{.}50$ in June if the market were less than $2,500$ while keeping their portfolio at $1\mathrm{.}25$ if the market were greater than $2,500.$

b$.$ Show in a table the proportional changes in the S&P $500$ in June from its March level of $2,500,$ the proportional changes in the portfolio, the values of the portfolio corresponding to the spot index, the call or put option values, the call or put position$$s cash flow, the option$-$hedged portfolio values, and the proportion changes in the option$-$hedged portfolio from its $$500$ million current value, proportion changes in the option$-$hedged portfolio value to the proportional changes in S&P $500.$ Evaluate for possible spot index values on the June expiration date of $1,750,2,000,2,250,2,500,2,750,3,000,$ and $3,250.$ You may use Excel and copy and paste $($pic$)$ your table in the exam.

c$.$ Suppose Malin decided that the cost of the options was too expensive and therefore decided to just lower her beta to $0\mathrm{.}5$ in June, irrespective of whether the market increased or decreased. Explain how Malin could do this with June S&P $500$ futures contracts that in March were at f$0=$ $$2,500($multiplier $=$ $$250).$