At $9$:$00$ am $10$th March $2024,$ your brokerage company receives an order from a client. This client want to buy a $1,000$ shares in Pear Inc $($PEAR$)$ and plans to hold these shares for $3$ months until $10$th June $2024.$ The midprice of PEAR is $$170\mathrm{.}73$ when you receive the order, and the client expects its midprice to be $$194\mathrm{.}50$ on $10$th June $2024.$ In this illiquid market for PEAR shares, your company can only fulfill $70\%$ of the total order size by $14$th March with an average price of $$179\mathrm{.}7.$

$1.$ Calculate the execution cost, opportunity cost and implementation shortfall.

$2.$ Let $12$th March be the date at which the buy order begins to be executed, at the midprice of $$174\mathrm{.}89.$ Calculate the delay portion of the execution component of the implementation shortfall.

$3.$ Consider another scenario where your company can fulfill $85\%$ of the total order size by $14$th March at an average price of $$180\mathrm{.}2.$ And another scenario where your company can fulfil $55\%$ of the total order size at an average price of $$179.$ Compare the implementation shortfalls in these three scenarios.