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

the investor's required rate of return is $13\mathrm{.}5$ percent,

the expected level of earnings at the end of this year $(E_1)$ is $$6,$

the retention ratio is $50$ percent,

the return on equity $($ROE$)$ is $15$ percent $($that is$,$ it can earn $15$ percent on reinvested earnings$),$ and

similar shares of stock sell at multiples of $8\mathrm{.}333$ 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 company increased its retention rate to $60$ percent $($holding all else constant$)?$ What would happen to the $\frac{P}{E}$ ratio $(\frac{P}{E_1})$ and stock price if the company paid out all its earnings in the form of dividends?

f$.$ What have you learned about the relationship between the retention rate and the $\frac{P}{E}$ ratios?