$1.$ Consider an all$-$equity financed firm, with $1$ million shares outstanding, a required return on

equity of $10\%,$ and expected earnings of $$10$ million$/$year$,$ in perpetuity, all of which will bepaid out to shareholders.

a$.$ Assume the $$10$ million$/$year in expected earnings are paid out as dividends.

i$.$ What is the expected dividend per share?

ii$.$ Assuming the first dividend payment occurs in $1$ year, i$.$e$.,$ at the end of the first

year, what is the equity value per share at time $0?($Hint: This is just an application

of the standard dividend discount model.$)$

iii. What is the total value of the equity $($in $mill.$)$ at time $0?$

b$.$ Under the same assumptions as part a:

i$.$ What is the expected equity value per share at time $1$ just before the dividend

payment, i$.$e$.,$ include the time $1$ dividend payment in this value at time $1($this is the

cum$-$dividend value$)?$

ii$.$ What is the expected equity value per share at time $1$ just after the dividend

payment, i$.$e$.,$ exclude the time $1$ dividend payment in this value at time $1($this is

the ex$-$dividend value$)?$

c$.$ The return on equity can be broken up into two parts, the capital gains return, i$.$e$.,$ the return attributable to price appreciation, and the dividend return, i$.$e$.,$ the returnattributable to dividend payments $($the dividend yield$).$

i$.$ If you buy the stock at time $0,$ and sell it at time $1$ just before the dividend payment

$($at the cum$-$dividend value$),$ what are the expected capital gains and dividend

returns?

$2$

ii$.$ If you buy the stock at time $0,$ and sell it at time $1$ just after the dividend payment

$($at the ex$-$dividend value$),$ what are the expected capital gains and dividend returns?

d$.$ Now assume that the firm decides to take the $$10$ million of expected earnings at time $1$

and use it to buy back equity. In every subsequent year, the $$10$ million of expected

earnings will again be paid out as dividends.

i$.$ At what price $($value$)$ per share should they expect to be able to buy back equity at

time $1?$

ii$.$ How many shares can they expect to buy back at this price $($in mill.$)?$

iii. How many shares can they expect to remain outstanding after this buyback $($in

mill.$)?$

iv$.$ What are the expected dividend per share going forward $($starting at time $2)?$

v$.$ What is the value per share of this expected dividend stream $($i$.$e$.,$ the per share

value of dividends starting at time $2$ and going on forever$)$ at time $1?$

vi$.$ What is the value per share of this same expected dividend stream $($i$.$e$.,$ the per

share value of dividends starting at time $2$ and going on forever$)$ at time $0?($Note

that this is the value at time $0$ of the expected dividends received by a shareholder

who does not participate in the share buyback at time $1.)$

e$.$ Finally, assume that the firm expects to use $$5$ million of earnings every year to buy back

equity and the remaining $$5$ million of earnings to pay out dividends. Assume that, at the end of each year, the buyback occurs first, and then dividends are paid to the remaining shareholders.

i$.$ At what price $($value$)$ per share should they expect to be able to buy back equity at

time $1?$

ii$.$ How many shares can they expect to buy back at time $1$ at this price $($in mill.$)?$

iii. How many shares can they expect to remain outstanding at time $1$ after this

buyback $($in mill.$)?$

iv$.$ What are expected dividends per share at time $1($on the remaining equity$)?$

v$.$ What is the expected ex$-$dividend value $($price$)$ per share at time $1?$

vi$.$ What is the expected cum$-$dividend value $($price$)$ per share at time $2?$

vii. How many shares can they expect to buy back at time $2$ at this price $($in mill.$)?$

viii. How many shares can they expect to remain outstanding at time $2$ after this

buyback $($in mill.$)?$

ix$.$ What are expected dividends per share at time $2?$

x$.$ At what annual rate do expected dividends grow from time $1$ to time $2?$

xi$.$ Assume that an equity holder at time $0$ expects to receive the time $1$ dividend per

share from part iv and expects dividends to grow in perpetuity at the growth rate

calculated in part x$.$ What is the value of this growing dividend stream at time $0?$

$($Note that this is the value at time $0$ of the expected dividends received by a

shareholder who does not participate in the share buybacks at any time.$)$