Show$-$Your$-$Work $2.$ You are a fund manager and manage a portfolio worth $$50$ million with a beta of $0\mathrm{.}8.$ You are concerned about the performance of the market over the next three months and plan to use three$-$month futures contracts on a well$-$diversified index to hedge your risk. The current index level is $1,250.$ The current three$-$month futures price is $1,259.$ One futures contract is on $250$ times the index. The risk$-$free rate is $1\mathrm{.}5\%$ per three months, and the dividend yield on the index is $0\mathrm{.}75\%$ per three months.

a$).$ What position should you take to hedge $($minimize$)$ exposure to the market over the next three months? Note: Specify

i$)$ whether you want to take a long or short position; and

ii$)$ how many contracts you want to buy or sell $($round the number of contracts to the nearest integer$).$

b$).$ What will be the rate of return on the index if the index in three months is $1,000?$

c$).$ Based on the CAPM and your results from part b$),$ what would be the rate of return on your portfolio? Note: the index return can be used as a proxy for the market return.

d$).$ What would be dollar gain$/$loss your portfolio based on your results from part c$)?$

e$).$ If the futures price in three months is $$1,007\mathrm{.}5,$ what would be the jollar gain$/$loss on one futures contract?

f$).$ Calculate the net dollar gain$/$loss of your hedging strategy over the three months based on the results from parts a$)$ through e$).$