Question $1(9$ marks$)$ Suppose the spot CAD$/$GBP exchange rate is $1\mathrm{.}6932,$ the $1-$year continuously compounded rate in Canada is $6\%.$ Suppose the $1-$year futures exchange rate is $1\mathrm{.}7231$

$($a$)$ What is the interest rate prevailing in the UK$?(3$ marks$)$

$($b$)$ What is the $6-$month futures exchange rate? $(3$ marks$)$

$($c$)$ Suppose the $6-$month futures exchange rate is $1\mathrm{.}9701.$ Is there an arbitrage? If so$,$ show how you can make a risk$-$free profit with a loan of $1$ million. $(3$ marks$)$

Question $2(8$ marks$)$ General Mills stock sells at $$44\mathrm{.}46$ and is expected to pay dividends of $$0\mathrm{.}6$ in $3,6,9$ and $12$ months respectively. The risk$-$free rate is $5\%$ per annum continuously compounded for all maturities. We consider the $1-$year futures contract on General Mills.

$($a$)$ What is the theoretical $1-$year futures price? $(3$ marks$)$

$($b$)$ The $1-$year futures market price is $$46.$ Is there an arbitrage? If so$,$ how can we benefit from it$?$ Show all details. $(4$ marks$)$

$($c$)$ Based on the futures market price in part $($b$),$ what is the value of a short futures contract on General Mills $6$ months from now if the futures price in $6$ months is $44$$$?(1$ mark$)$

Question $3(6$ marks$)$ Gold sells at $$1,200$ per ounce. It has a convenience yield denoted by y and a storage cost u $=5\%($per annum continuously compounded$).$ The risk$-$free interest rate is $6\%$ per annum continuously compounded. The $6-$month futures price is $$1,247\mathrm{.}101.$ A hedge fund takes $70$ long positions in the $1-$year futures contract on gold today. Each contract is on $100$ ounces

$($a$)$ What is the convenience yield? $(2$ marks$)$

$($b$)$ Suppose that the hedge fund closes out its positions $9$ months from now and makes a total gain of $$70,000.$ What is the spot price of gold $9$ months from now if there is no arbitrage? $(4$ marks$)$

Question $4(7$ marks$)$ The S&P $500$ spot is $2,929\mathrm{.}67$ and it is expected to pay a dividend yield of $3\%.$ The risk$-$free rate is $4\%$ per annum continuously compounded. Each contract is on $$250$ times the futures price.

$($a$)$ What is the theoretical $6-$month futures price? $(3$ marks$)$

$($b$)$ The $6-$month futures price is $3,000.$ Is there an arbitrage? If so$,$ show how we can benefit from it$?$ Show all details. $(4$ marks$)$