$5.$ It is November $5.$ A company has a portfolio of stocks worth $$145$ million. The beta of the portfolio is $1\mathrm{.}72.$ The company would like to use the CME December futures contract on the S&P $500$ to change the beta of the portfolio to $0\mathrm{.}43$ during the period from November $5$ to December $10.$ The December S&P $500$ index futures price is currently $2,590,$ and each contract is on $$250$ times the index.

$($a$)$ What position should the company take? How many December S&P $500$ futures contracts should the company buy or sell now?

$($b$)$ Suppose that the company changes its mind and decides to increase the beta of the portfolio from $1\mathrm{.}72$ to $1\mathrm{.}96.$ What position in futures contracts should it take? How many December S&P $500$ futures contracts should the company buy or sell now?

$6.$ The cash prices of six$-$month and one$-$year Treasury bills are $98\mathrm{.}0$ and $95\mathrm{.}0,$ respectively. A $1\mathrm{.}5-$year bond that will pay coupons of $$3$ every six months currently sells for $$96.$ A two$-$year bond that will pay coupons of $$7$ every six months currently sells for $$101.$ Assume the principal of the bond is $$100.$

Calculate the $($a$)$ six$-$month, $($b$)$ one$-$year, $($c$)1\mathrm{.}5-$year, and $($d$)$ two$-$year zero rates.

$7.$ A fund manager has a portfolio worth $$95$ million with a beta of $1\mathrm{.}16.$ The manager is concerned about the performance of the market over the next two months and plans to use three$-$month futures contracts on the S&P $500$ to hedge the risk.

The current level of the S&P $500$ index is $2,675,$ the risk$-$free rate is $2\%$ per annum, and the dividend yield on the index is $4\%$ per annum. The current $3-$month S&P $500$ index futures price is $2,660,$ and one contract is on $250$ times the index.

$($a$)$ What position should the fund manager take to eliminate all risk exposure to the market over the next two months? $($B$\text{'}=0)$ How many three$-$month S&P $500$ futures contracts should the fund manager buy or sell now? $($Rounding to the nearest whole number.$)$

$($b$)$ Calculate the effect of your strategy on the fund manager's returns if the level of the S&P $500$ index in two months is $2,585.$ Assume that the $3-$month S&P $500$ index futures price is $2,570$ in two months.

What would be the expected value of the fund manager's hedged portfolio $($including the spot and futures positions$)$ in two months?

$8.$ Suppose that the $3-$year and $3\mathrm{.}5-$year zero rates with continuous compounding are $2\mathrm{.}5\%$ and $2\mathrm{.}55\%,$ respectively.

$($a$)$ What is the forward rate for the six$-$month period beginning in $3$ years $(3$R$35)($from Year $3$ to Year $3\mathrm{.}5)$ with continuous compounding?

$($b$)$ What is the forward rate for the six$-$month period beginning in $3$ years $(3$R$3\mathrm{.}5)($from Year $3$ to Year $3\mathrm{.}5)$ with semiannual compounding?

$($c$)$ What is the $($Year $0)$ value of an FRA that promises to pay the lender $2\mathrm{.}75\%($compounded semiannually$)$ on a principal of $$3$ million for the six$-$month period starting in $3$ years $($from Year $3)$

$9.$ At the end of one day $($t$=1),$ a clearinghouse member is long $90$ futures contracts, and the settlement price is $$650$ per contract. On the following day $($t$=2),$ the member becomes responsible for clearing additional new $25$ short futures contracts $($day trades$),$ entered into at a price of $$685$ per contract. The settlement price at the end of this day is $$665.$ The initial margin is $$2,200$ per contract. How much does the member still have to add to his margin account with the exchange clearinghouse $($on t$=2)$ to meet the initial margin requirement for the $25$ new futures contracts, after considering the gain$/$loss of the $90$ old futures contracts and the $25$ new futures contracts?

$10.$ The standard deviation of monthly changes in the spot prices $($in dollars per barrel$)$ of crude oil is $0\mathrm{.}096.$ The standard deviation of monthly changes in the futures prices of crude oil for the closest contract is $0\mathrm{.}087.$ The correlation between the futures price changes and the spot price changes is $0\mathrm{.}92.$ It is now April $15.$ A oil producer is committed to purchasing $65,000$ barrels of crude oil on December $15.$ The producer wants to use the December crude oil futures contracts to hedge its risk. Each contract is for the delivery of $1,000$ barrels of crude oil.

What strategy should the oil producer follow? How many December crude oil futures contracts should the oil producer sell or buy now?