Both Home Away Inn Ltd$.($HAI$)$ and Budget Inns Ltd$.($BIL$)$ operate in the fast growing business of budget accommodation. They are both all$-$equity firms and have market values $($as stand$-$alone firms$)$ of $$200$ million and $$80$ million, respectively. The Board of HAI believes that acquiring BIL will bring a synergy of $$70$ million due to complementary locations of the two firms. The Board of BIL has indicated that it will sell BIL if it is offered $$120$ million in cash.

$($a$)$ What is the value of HAI after the acquisition if it pays $$120$ million in cash?

$($b$)$ What is the NPV of the acquisition to HAI?

$($c$)$ If investors estimate a probability of $60\%$ that the merger will take place, what will be the market value of HAI after it makes the offer but before the merger is completed?

$($d$)$ Will the manager of HAI overestimate or underestimate the NPV of the acquisition to HAI if the market value obtained from part $($c$)$ is used?