ABC is a U$.$S$.$ firm that holds some stocks in the United Kingdom and faces three different scenarios in one year's time: With $50\%$ probability, the spot exchange rate will be $$\frac{2\mathrm{.}00}{}$ and the pound price of the asset will be $2,500$; with $25\%$ probability, the spot exchange rate will be $$\frac{2\mathrm{.}20}{}$ and the pound price of the asset will be $2,000$; with $25\%$ probability, the spot exchange rate will be $$\frac{1\mathrm{.}80}{}$ and the pound price of the asset will be $3,000.$

$($a$)(2$ points$)$ What is the mean value $($in US dollars$)$ of the British stocks?

$($b$)(3$ points$)$ What is the covariance between the US dollar value of the British stocks and the spot $\frac{\text{\$}}{}$ exchange rate?

$($c$)$ points$)$ To fully hedge the exchange rate exposure of the British stocks, should the US firm buy or sell pounds in the forward market? And what is the amount in pounds the US firm should buy or sell in the forward market?

$($d$)(3$ points$)$ What is the percentage of the total variance of the British stocks $($in US dollars$)$ that cannot be removed by hedging at the amount found in part $($c$)?$