Problem $15$: Investment mechanism and expected returns$/$risks $(21\%)$

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. If letting you make a trading decision, do you think you should buy the stock or short sell the stock? Explain why.$($Do not round intermediate calculations. Round your answers to $4$ decimal places.$)$

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$)$

C$.$ Base on your trading decision and the conditions in A$,$ you start to submit a market order. Suppose you decide to trade $1500$ shares of TSLA and not use any margin. You find the current order book structure is as the follows:

Limit Buy Orders$($Bid$)$ Limit Sell Orders$($Ask$)$

Price $($Per Share$)$ Shares Price $($Per Share$)$ Shares

$ $179\mathrm{.}5200$ $ $181\mathrm{.}5100$

$ $179\mathrm{.}2500$ $ $181\mathrm{.}6100$

$ $178\mathrm{.}8700$ $ $182\mathrm{.}2300$

$ $178\mathrm{.}3400$ $ $182\mathrm{.}51000$

$ $178\mathrm{.}2600$ $ $182\mathrm{.}8200$

On Aug $13$th$,2024,$ you closed your position by submitting a limit order of $$173$ per share. In addition, the dividend paid during the two$-$month period is $$2$ per share. The order was successfully executed. What is your profit or loss $($not return$)$ after you closed the position?