You are the manager of a stock portfolio worth $$10,500,000.$ It has a beta of $1\mathrm{.}15.$ During the next three months, you expect a correction in the market that will take the market down about $5$ percent; thus, your portfolio is expected to fall about $5\mathrm{.}75$ percent $(5$ percent times a beta of $1\mathrm{.}15).$ You wish to lower the beta to $1.$ An S&P$500$ futures contract with the appropriate expiration is priced at $415\mathrm{.}75$ with a multiplier of $$500.$

a$.$ Should you buy or sell futures? How many contracts should you use?

b$.$ In three months, the portfolio has fallen in value to $$9,870,000.$ The futures has fallen to $402\mathrm{.}35.$ Determine the profit and portfolio return over the quarter. How close did you come to the desired result?

c$.$ Repeat questions $a.$ and $b.$ when you want to lower the beta to $0$