Answer with shown work

$4.$ Assume the current Price$-$to$-$Earnings $($P$/$E$)$ ratio for Firm A is $20,$ which is the same

as the S&P $500$ index. If Firm A is an average firm, what is the required rate of return?

Additionally, if Firm A is not an average firm, and its current stock price is $$100,$ with

forward earnings per share of $$5,$ a required rate of return of $10\%,$ and a plowback ratio

of $50\%,$ what is the expected Return on Equity $($ROE$)$ for Firm A$?$

$5.$ Suppose a mature firm decides to invest in a new project with an expected Return on

Equity $($ROE$)$ of $10\%.$ To fund this project, the firm reduces its dividend payment by

$50\%.$ The forward earnings are $$10$ per share, and the required rate of return is $8\%.$

What is the present value of the growth opportunity $($PVGO$)?$