Pat Plummets opens an account to sell short Crazy Cryptocurrency Exchange $($CCE$).$ Pat Plummets then shorts $2000$ shares of CCE at a price per share of $$80.$ The initial margin requirement is $50\%.$ The broker charges a borrowing fee of $5\%$ per year on the initial value of securities borrowed. No interest is earned on the deposits in the account. The broker requires a $30\%$ maintenance margin.

a$.$ How much does Pat Plummets need to deposit in the account initially?

b$.$ If in one year's time, CCE's stock price has increased to $$110$ and paid a dividend of $$1\mathrm{.}50$ per share. What is the remaining margin in the account? If Pat closes the position, what rate of return will Pat receive?

c$.$ If in one year's time, CCE's stock price has decreased to $$50$ and paid a dividend of $$1\mathrm{.}50$ per share. What is the remaining margin in the account? If Pat closes the position, what rate of retum will Pat receive?

d$.$ At what price will Pat receive a margin call if the stock price moves immediately afer putting on the position?

e$.$ At what price will Pat receive a margin call if the stock price moves in one ycar affer the interest and dividend?

f$.$ When CCE's stock actually falls to $$2$ in one year's time after having paid a dividend of $$1\mathrm{.}50$ per share, what rate of return will Pat receive?