Question $3.$

$($a$)$ As $2019$ came to an end, Carlson Corporation paid a quarterly dividend of $$1\mathrm{.}12$ per share. At that point in time Carlson$$s dividends were expected to grow by $1\%$ per quarter in perpetuity. Assuming a discount rate of $2\mathrm{.}0\%$ per quarter and that each dividend is paid at quarter end, what was the value of a Carlson share just after it paid that dividend?

$($b$)$ The emergence of Covid$-19$ led to severe cash flow problems at Carlson, and it announced in early $2020$ that it would suspend its dividend for the remainder of the year. As of the end of $2020,$ Carlson projected that it would not pay a dividend in either the first or the second quarter of $2021,$ but that it would pay a dividend of $$0\mathrm{.}25$ per share at the end of each of the third and fourth quarters of $2021.$ Further, Carlson projected that it would pay a dividend of $$0\mathrm{.}45$ per share at the end of each of the four quarters during $2022,$ and that it would be able to return its dividend to the $$1\mathrm{.}12$ level by the first quarter of $2023,$ after which it would be able to increase subsequent dividends by $1\%$ per quarter in perpetuity. Still assuming a discount rate of $2\mathrm{.}0\%$ per quarter, what was the value of a Carlson share at the end of $2020?$