$2.$ A firm's expected earnings in one year is $$5$ per share and the firm is going to invest $50\%$ of its earnings in an new investment opportunity and pays out the rest $50\%$ as dividends every year. Assume that once an investment is made, it will continue to generate a fixed rate of return every year going forward. Suppose that investors believe that the new investment opportunities the firm has in future years can generate a return of $10\%$ per year and investors demand a $10\%$ expected return to invest in this firm given the risk of these investments' cash flows. What should be the price per share? Suppose that the firm makes an announcement today that makes investors believe that the firms' investment opportunities can generate a return of $14\%$ per year, assuming that the plowback ratio and the required rate of return stays the same, how much will the price increase $($in percentage terms$)$ when the announcement is made? Does this news change the stock's expected return going forward $($after the price has reacted to the news$)?$