Solutions to Practice Problems – Chapter 16
1 a) Total value of equity = 6,000 + 28,000 = 34,000.
Stock price = 34,000/1,200 = 28.33
Stockholder wealth = value of
equity = $34,000
b)
Stock price before the dividend = 28.33
Dividend per share =
6,000/1,200 = 5.00
Ex-dividend stock price =
28.33 – 5 = 23.33
Wealth of s/h will not
change. Wealth after dividend =
dividend received + value of shares = 6,000 + 1,200*23.33 = 6,000 + 28,000 =
34,000
c) Stock price before the
dividend stays the same, $28.33
To pay an additional $3,000
as dividend, the firm must issue new shares worth $3,000.
Let N = the number of new shares issued.
After the firm has raised
$3,000 in new equity and paid the $9,000 dividend, it is left with assets worth
$28,000. So 28,000 is the total
value of the 1200 + N shares now outstanding.
The N new shares are worth $3,000.
=> Value of original 1,200
shares = 25,000
=> Ex-dividend stock price
= 25,000/1,200 = 20.83
The new shares are issued at
this ex-dividend price. Number of
shares issued = 3,000/20.83 = 144
Wealth of the original
stockholders after the dividend = dividend received + value of shares = 9,000 +
1,200*20.83 = 9,000 + 25,000 = 34,000. Once
again, stockholder wealth is not affected by the change.
2 a) Before the dividend, the
existing assets of the firm are worth 1,200 + 22,000 = 23,200.
The value of the investment
opportunity = the NPV of the project = 1,000
The total value of equity =
23,200 + 1,000 = 24,200.
Stock price = 24,200/4,000 =
6.05
The firm proposes to pay out
1 * 4000 = 4,000 in dividends. In
the context of this problem, the sources and uses of cash, will give us:
Cash + New Equity = Investment +
Dividends
1,200 + New Equity = 3,500 + 4,000
=> New Equity = $6,300
Once the firm has issued the new shares, paid the dividend
and invested in the project, it has old assets worth 22,000 and new assets worth
3500 + 1000 = 4,500, for a total value of $26,500.
(The PV of the assets acquired for the project = Investment + NPV.)
There are now 4000 + N shares outstanding.
The N new shares are worth 6,300, so the original 4000 shares are worth
20,200, for a stock price of 20,200/4,000 = 5.05.
The original wealth of stockholders = value of their 4,000 shares =
24,200.
The final wealth of stockholders = dividend received + value of their
4,000 shares = 4,000 + 20,200 = 24,200
b) N new shares are
issued at the ex-dividend price of $5.05 to raise $6,300. The number of new
shares issued = 6,300/5.05 = 1,247.53
3 a)
Before the dividend, the existing assets of the firm are worth 1,500 + 7,750 =
9,250.
The value of the investment
opportunities = the total NPV of the projects = 425 + 325 = 750
Cum-dividend value of equity
= 9,250 + 750 = 10,000.
Cum-dividend stock price =
10,000/1,000 = 10
The firm proposes to pay out
2 * 1000 = 2,000 in dividends. In
the context of this problem, the sources and uses of cash, will give us:
Cash + New Equity = Investment +
Dividends
1,500 + New Equity = 1,800 + 2,000
=> New Equity = $2,300
Once the firm has issued the new shares, paid the dividend
and invested in the project, it has old assets worth 7,750 and new assets worth
1425 + 1125 = 2,550, for a total value of $10,300.
There are now 1000 + N shares outstanding.
The N new shares are worth 2,300, so the original 1000 shares are worth
8,000 for a stock price of $8.
=> Ex-dividend value of
equity = 10,300
Ex-dividend stock price = 8
N new shares are issued at the ex-dividend price of $8 to raise $2,300.
The number of new shares issued = 2,300/8 = 287.50
The original wealth of stockholders = value of their 1,000 shares =
10,000.
The final wealth of stockholders = dividend received + value of their
1,000 shares = 2,000 + 8,000 = 10,000
The fraction of the shares held by the original stockholders at the end
= 1000/1287.50 = 77.67%
b) With a stock
repurchase, the stock price doesn’t change.
The initial and final stock price will be $10.
Initially there are 1,000 shares outstanding with a total value of
$10,000.
They repurchase 200 old shares for $2,000. They issue 230 new shares for $2,300.
The total value of the firm is now $10,300 (just as it was for firm D),
and there are 1000 – 200 + 230 = 1,030 shares outstanding.
The initial wealth of the
original stockholders was 1000*10 = 10,000.
At the end the stockholders
who sold their shares have $2,000 in cash.
The stockholders who kept their shares have $8,000 worth of equity.
Their total wealth is still the same: 2,000 + 8,000 = 10,000
The fraction of the shares held by the original stockholders at the end
= 800/1030 = 77.67%
The number of shares repurchased and the number of new shares issued will both increase. However, they will increase by the same amount, so there will still be 1,030 shares worth $10 each at the end. The original stockholders will be left holding a smaller fraction of these 1030 shares.
4
a) Cum-dividend value of equity = total
value of all 10,000 shares = 8,000 + 240,000 + 900 = 248,900.
Cum-dividend
stock price = 248,900/10,000 = 24.89
Value
of new equity to be issued = Investment + dividend – cash = 5,000 + 7,500 –
8,000 = $4,500
Ex-dividend
value of equity = value of firm’s assets after
it has new equity and paid the dividend =
value of project + value of other assets = 5,900 + 240,000 = 245,900.
This
is the value of the original 10,000 shares plus the N new shares.
The N new shares are worth $4,500. So
the original 10,000 shares are worth 245,900 – 4,500 = 241,400.
Ex-dividend
stock price = 241,400/10,000 = 24.14
Number
of new shares issued = 4,500/24.14 = 186.41
Cum-dividend
stockholder wealth = original value of equity = $248,900
Ex-dividend
stockholder wealth = dividend + final value of original 10,000 shares = 7,500 +
10,000*24.14 = 248,900
The
fraction of the shares held by the original stockholders at the end =
10,000/(10,000 + 186.41) = 98.17%
b)
With a share repurchase, the stock price doesn’t change.
Initial and final stock price are both $24.89.
The
firm spends $8,000 to buy back shares at 24.89 per shares, so number of new
shares repurchased = 8,000/24.89 = 321.41
Value
of new equity to be issued = Investment + repurchase – cash = 5,000 + 8,000
– 8,000 = $5,000
Number
of new shares issued = 5,000/24.89 = 200.88
Initial
value of equity = 8,000 + 240,000 + 900 =
248,900 (same as with dividends)
Ex-dividend
value of equity = 5,900 + 240,000 = 245,900 (same as with dividends)
Initial
wealth of original s/h = 10,000 * 24.89 = 248,900
Final
wealth of original s/h = 8,000 + (10,000 – 321.41) * 24.89 = 248,900
Change
in wealth of original stockholders = 0
The
fraction of the shares held by the original stockholders at the end = (10,000
– 321.41)/( 10,000 – 321.41 + 200.88) = 97.97%
5
a) It
is not possible to make a pure dividend policy change; when you try to change
the dividend keeping the investment decision constant, the firm’s capital
structure necessarily changes. F When you
try to make a pure capital structure change, then dividend policy necessarily
changes, but not vice versa.
b)
When a firm increases its dividend, stockholders are receiving a higher
return; hence the firm’s cost of equity increases. F
The cost of equity is based on the expected return or required return.
When a firm increases its dividend, the stock price goes up and stockholders end
up with a higher actual return, but that has nothing to do with cost of equity.
c)
The reluctance of managers to cut dividends leads to a situation where
dividends reflect nor just current earnings but expected future earnings as
well; the market then punishes you for cutting dividends by reducing your stock
price.
d)
Dividends are typically increased when earnings go up and managers are
confident that earnings will remain at this higher level.
e)
An earnings announcement tells the market whether earnings went up or
down; the accompanying dividend announcement clarifies whether the earnings
change is temporary or permanent.
f)
In perfect capital markets, when a firm pays a dividend, the stock price
falls by exactly the amount of the dividend.
g)
If a firm is paying out $2 million to its stockholders, the stock price
will fall by the same amount whether the $2 million is paid out through a
dividend or a stock repurchase.
h)
In PCM, stockholders who do not need cash from their portfolios may not
care whether a firm pays dividends or not, but stockholders who need cash from
their portfolio for living expenses prefer dividend paying firms.
i)
Corporations do not pay taxes on their dividend income, so they are
indifferent between dividend income and capital gains income. F
That's tax-exempt institutions like mutual funds and pension funds.
Corporations do pay taxes on dividend income (though only on 30% of the dividend
amount, since 70% is tax-exempt).
j)
In perfect capital markets, stockholders are indifferent between buying
shares in the unlevered firm or borrowing
money to buy shares in an otherwise identical levered firm. F
They are indifferent between buying shares in the levered
firm or borrowing money to buy shares in an otherwise identical unlevered
firm.