Suppose common shares of Company A are trading at $$1687\mathrm{.}00$ per share. Company historical dividend yield is $4\mathrm{.}0$ percent per annum based on expected future price at the end of $12$ months period from today. Company recorded capital gains over the past $12$ months of $11\mathrm{.}0\%.$

a$)$ Assume that over the next $12$ months, the share price is expected to fall by the same percentage rate as it gained in prior $12$ months. What are the price investors will be willing to pay for this stock today, if the discount rate is $5$ percent per annum? Is the stock over$-$ or under$-$valued at $$1687\mathrm{.}00?$ How much is it over$-$ or under$-$valued by$?$

b$)$ Now, suppose the discount rate that applies to equity markets falls to $2\mathrm{.}5$ percent, but dividend yield for the next twelve months rises to $7\mathrm{.}5\%.$ Is the stock over$-$ or under$-$valued at $$1687\mathrm{.}00?$ How severely?

c$)$ What are the nominal and real rates of return to be earned from the stock in $($a$)$ and $($b$)$ above?

d$)$ How sensitive is over$-$ or under$-$valuation in part $($a$)$ to the assumed discount rate? Hint $1$: can vary discount rate to see how over$-/$ under$-$valuations changes, while holding all other assumptions constant as in $($a$).$ Hint $2$: visualizing your over$-/$under$-$valuation estimates $($Y$-$axis$)$ as a function of assumed discount rates $($X$-$axis$)$ will be super helpful here.