$($Common stock valuation$)$ Assume the following:

the investor's required rate of return is $18$ percent,

the expected level of earnings at the end of this year $(E_1)$ is $$10,$

the retention ratio is $40$ percent,

the return on equity $($ROE$)$ is $18$ percent $($that is$,$ it can earn $18$ percent on reinvested earnings$),$ and

similar shares of stock sell at multiples of $5\mathrm{.}556$ times earnings per share.

Questions:

a$.$ Determine the expected growth rate for dividends.

b$.$ Determine the price earnings ratio $(\frac{P}{E_1}).$

c$.$ What is the stock price using the P$/$E ratio valuation method?

d$.$ What is the stock price using the dividend discount model?

e$.$ What would happen to the $\frac{P}{E}$ ratio $(\frac{P}{E_1})$ and stock price if the firm could earn $23$ percent on reinvested earnings $($ROE$)?$

f$.$ What does this tell you about the relationship between the rate the firm can earn on reinvested earnings and $\frac{P}{E}$ ratios?