You want to sell short $100$ shares of company XYZ because you expect the price of XYZ to drop in the near future. The current price of XYZ is $$46\mathrm{.}5,$ and you are required to post a $50\%$ margin in cash. For simplicity, let$$s ignore fees, interests, and other charges from the broker.

$1.$ Construct a stylized balance sheet with assets on a side, and liabilities and equity on the other side. Compute leverage.

$2.$ Now assume that the price of XYZ$,$ contrarily to your expectation, goes up to $$55.$ Construct the newbalance sheet and compute leverage.

$3.$ Your broker requires you to keep a minimum amount of cash in your account, or it will issue a margin call. This minimum amount is equal to the value of your short position $($based on the most recent price of XYZ$)$ plus a $30\%$ maintenance margin. What is the minimum amount of money that you should add to your account to keep the margin above the maintenance margin $($i$.$e$.,$ to avoid a margin call$)?$ What is the threshold for leverage such that, if leverage increases above the threshold, a margin call is issued by the broker?