XYZ Inc. has expected earnings over the next year of $$\frac{3}{?}$ share $)=(3.$ The company is expected to maintain an earnings retention rate of $40\%,$ i$.$e$.,60\%$ of earnings are expected to be paid out as dividends every year. The company has a beta of $1\mathrm{.}5,$ the risk$-$free rate is $4\%,$ and the market risk premium is also $5\%.$

a$.$ If the growth rate in earnings is expected to be $4\%$ in perpetuity

i$.$ What is the value of the stock?

ii$.$ What is the expected price a year from now?

iii. What is the expected holding period return over the next year? How much of this return is due to capital gains $($price appreciation$)$ and how much is attributable to dividend yield?

iv$.$ What ROE justifies this growth rate?

v$.$ What is the present value of growth opportunities for this stock?

b$.$ If the ROE is expected to be $15\%$ in perpetuity

i$.$ What is the implied growth rate?

ii$.$ What is the value of the stock?

iii. What is the present value of growth opportunities?

c$.$ If the current price of the stock is $$\frac{27}{?}$ share

i$.$ What is the implied growth rate?

ii$.$ What is the implied ROE?

iii. What is the present value of growth opportunities?