The stock of TSLA sells $$180$ per share $($Latest Close Price$)$ on June $13$th$,2024.$ The rate on Treasury bills is $1\mathrm{.}4\%.$ Suppose an analyst at Morgan Stanley just announced the following forecast on the TSLA stock price on Aug $13$th$,2024$ and dividend payment before Aug $13$th$,2024$:

Economy Status Probability TSLA Stock Price Per Share Dividend Payment Per Share

Boom $20\%$ $$205$ $$3$

Normal economy $30\%$ $$183$ $$2$

Recession $50\%$ $$159$ $$1$

A$.$ Suppose you totally trust this forecast from the analyst, calculate the expected holding$-$period return and standard deviation of the holding$-$period return over one month for APPL stock as well as the corresponding Sharpe ratios. Please do the Sharpe Ratios

B$.$ Based on your trading decision and the conditions in A$,$ you traded some shares of TSLA on June $13$th$,2024.$ Since you do not have enough fund, you need to use margin. Suppose the maintenance margin is $30\%.$ On Aug $13$th$,2024,$ TSLA price reaches the worst case $($according to the analyst$$s forecast$)$ and you receive a margin call. What is the possible range of initial margin required by the broker? $($Hint: Think about the meaning of worst case which depends on your trade decision$)$