$6$

The following table gives data on monthly changes in the spot price and the futures price for a certain commodity. Use the data to calculate a minimum variance hedge ratio.

$\backslash$table$[[$Spot Price Change,$+0\mathrm{.}50,+0\mathrm{.}61,-0\mathrm{.}22,-0\mathrm{.}35,+0\mathrm{.}79],[$Futures Price Change,$+0\mathrm{.}56,+0\mathrm{.}63,-0\mathrm{.}12,-0\mathrm{.}44,+0\mathrm{.}60],[,,,,,],[$Spot Price Change,$+0\mathrm{.}04,+0\mathrm{.}15,+0\mathrm{.}70,-0\mathrm{.}51,-0\mathrm{.}41],[$Futures Price Change,$-0\mathrm{.}06,+0\mathrm{.}01,+0\mathrm{.}80,-0\mathrm{.}56,-0\mathrm{.}46]]$

$7$

It is July $16.$ A company has a portfolio of stocks worth $$100$ million. The beta of the portfolio is $1\mathrm{.}2.$ The company would like to use the December futures contract on a stock index to change beta of the portfolio to $0\mathrm{.}5$ during the period July $16$ to November $16.$ The index is currently $1,000,$ and each contract is on $$250$ times the index.

a$)$ What position should the company take?

b$)$ Suppose that the company changes its mind and decides to increase the beta of the portfolio from $1\mathrm{.}2$ to $1\mathrm{.}5.$ What position in futures contracts should it take?

$8$

A fund manager has a portfolio worth $$50$ million with a beta of $0\mathrm{.}87.$ 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 $S$&$P500$ to hedge the risk. The current level of the index is $1250,$ one contract is on $250$ times the index, the risk$-$free rate is $6\%$ per annum, and the dividend yield on the index is $3\%$ per annum. The current $3$ month futures price is $1259.$

a$)$ What position should the fund manager take to eliminate all exposure to the market over the next two months?

b$)$ Calculate the effect of your strategy on the fund manager's returns if the level of the market in two months is $1,000,1,100,1,200,1,300,$ and $1,400.$ Assume that the onemonth futures price is $0\mathrm{.}25\%$ higher than the index level at this time.