Your firm has $10$ million shares outstanding, and you are about to issue $5$ million new shares in an IPO. The IPO price has been set at $$20$ per share, and the underwriting spread is $7\%.$ The IPO is a big success with investors, and the share price rises to $$50$ the first day of trading.

a$.$ How much did your firm raise from the IPO?

b$.$ What is the market value of the firm after the IPO?

c$.$ Assume that the post$-$IPO value of your firm is its fair market value. Suppose your firm could have issued shares directly to investors at their fair market value, in a perfect market with no underwriting spread and no underpricing. What would the share price have been in this case, if you raise the same amount as in part $($a$)?$

d$.$ Comparing part $($b$)$ and part $($c$),$ what is the total cost to the firm's original investors due to market imperfections from the IPO?