# Finance 351_ Corporate Finance_ Problem Set 7 - Duke University s

Document Sample

```					                                        DUKE UNIVERSITY

FINANCE 351 - CORPORATE FINANCE
Problem Set #7
Prof. Simon Gervais                                                             Fall 2011 – Term 2

Questions

1. Suppose the corporate tax rate is 40%, and investors pay a tax rate of 15% on income from
dividends or capital gains and a tax rate of 1 (i.e., 33.3%) on interest income. Your ﬁrm
3
decides to add debt so it will pay an additional \$15 million in interest each year. It will
pay this interest expense by cutting its dividend (i.e., the operations will not change, and so

(a) How much will the debt-holders receive after paying taxes on the interest they earn each
year?
(b) By how much will the ﬁrm need to cut its dividend each year to pay this interest expense?
(c) By how much will this cut in the dividend reduce equity-holders’ annual after-tax income
(d) How much less will the government receive in total revenues each year?
(e) What is the eﬀective tax advantage of debt t∗ ?

2. Suppose that, in an eﬀort to reduce the federal deﬁcit, Congress increases the top personal
tax rate on interest and dividends to 44% but retains a 20% tax rate on realized capital gains.
The corporate tax rate stays at 35%. Assuming that capital gains are half of equity income,
compute the total corporate plus personal taxes paid on debt versus equity income if

(a) all capital gains are realized immediately;
(b) capital gains are deferred forever.

3. The XYZ Co. is assessing its current capital structure and its implications for the welfare of
its security holders. XYZ is currently ﬁnanced entirely with common stock, of which 1,000
shares are outstanding. Given the risk of the underlying cash ﬂows (EBIT, earnings before
interest and taxes) generated by XYZ, investors currently require a 20% return on the XYZ
common.
The company pays out all expected earnings as dividends to common stockholders, and these
expected earnings are based on the expected operating earnings (EBIT) generated by the
ﬁrm’s assets. XYZ estimates that operating earnings may be either 1,000, 2,000 or 4,200 with
respective probabilities of 0.1, 0.4 and 0.5 depending on future economic conditions. Further,
the ﬁrm expects to produce a level stream of EBIT in perpetuity (assume that, every year,
Depreciation is equal to CapEx and that ∆NWC = 0). Assume that the corporate and
personal tax rates are equal to zero.

(a) Given the above facts, compute
(i) the value of the ﬁrm;

1
(ii) the market value of a common share;
(iii) the expected earnings per share of common stock;
(iv) the ﬁrm’s weighted average cost of capital.
(b) The president of XYZ has come to the conclusion that shareholders would be better oﬀ
if the company had equal proportions of debt and equity. He therefore proposes to issue
\$7,500 of debt at an interest rate of 10% and use the proceeds to repurchase equity.
Analyze this proposition by computing
(i)   the   new value of the ﬁrm;
(ii)   the   value of debt;
(iii)   the   value of equity;
(iv)    the   number of common shares repurchased, and the price of one such share;
(v)    the   required rate of return on equity;
(vi)    the   ﬁrm’s weighted average cost of capital.
(c) Lift the assumption of no corporate taxes and now assume that the corporate income
tax rate is 40%. Assuming no risk of bankruptcy for a debt-equity ratio less than one,
how does the existence of a corporate income tax aﬀect your answers to part (b)? If
bankruptcy risks were a concern, how would this (qualitatively) aﬀect your answers?
(d) Now lift the assumption that personal taxes are zero. Suppose that returns to equity
holders are eﬀectively not taxed but that interest is taxed at the personal level. How
will your answer to parts (c.i) to (c.iii) change if the recipients of interest on the ﬁrm’s
debt are in the
(i) 30% personal income tax bracket?
(ii) 40% personal income tax bracket?

4. Gladstone Corporation is about to launch a new product. Depending on the success of the
new product, Gladstone may have one of four values next year: \$150 million, \$135 million,
\$95 million, or \$80 million. These outcomes are all equally likely (i.e., they occur with a 25%
probability each), and this risk is diversiﬁable. Suppose that the riskfree interest rate is 5%
and assume perfect capital markets (i.e., no market imperfections).

(a) What is the initial value of Gladstone’s equity without leverage?

Now suppose that Gladstone has zero-coupon debt with a \$100 million face value due next
year.

(b) What is the initial value of Gladstone’s debt?
(c) What is the yield-to-maturity of the debt? What is its expected return?
(d) What is the initial value of Gladstone’s equity? What is Gladstone’s total value with
debt?

5. As in the last problem, Gladstone Corporation is about to launch a new product. Depending
on the success of the new product, Gladstone may have one of four values next year: \$150
million, \$135 million, \$95 million, or \$80 million. These outcomes are all equally likely (i.e.,
they occur with a 25% probability each), and this risk is diversiﬁable. Suppose that the

2
riskfree interest rate is 5% and that, in the event of default, 25% of the value of Gladstone’s
assets will be lost to bankruptcy costs. Ignore all other market imperfections, such as taxes.

(a) What is the initial value of Gladstone’s equity without leverage?

Now suppose that Gladstone has zero-coupon debt with a \$100 million face value due next
year.

(b) What is the initial value of Gladstone’s debt?
(c) What is the yield-to-maturity of the debt? What is its expected return?
(d) What is the initial value of Gladstone’s equity? What is Gladstone’s total value with
leverage? Suppose that Gladstone has 10 million shares outstanding and no debt at the
start of the year.
(e) If Gladstone does not issue debt, what is its share price?
(f) If Gladstone issues debt of \$100 million due next year and uses the proceeds to repurchase
shares, what will its share price be? Why does your answer diﬀer from that in part (e)?

6. Nadir Inc., an unlevered ﬁrm, has expected earnings before interest and taxes of \$2 million
per year. Nadir’s tax rate is 40%, and its market value is V = E = \$12 million. The stock
has a beta of 1, the riskfree rate is 9%, and assume that rm − rf = 6%.
Management is considering the use of debt; debt would be issued and used to buy back
stock, and the size of the ﬁrm would remain constant. The default-free interest rate on debt
is 12%. Because interest expenses are tax deductible, the value of the ﬁrm would tend to
increase as debt is added to the capital structure, but there would be an oﬀset in the form of
the rising cost of bankruptcy. The ﬁrm’s analysts have estimated that the present value of
any bankruptcy cost is \$8 million, and that the probability of bankruptcy will increase with
leverage according to the following schedule:

Value of        Probability of
Debt            Failure (%)
2,500,000            0.0
5,000,000            8.0
7,500,000           20.5
8,000,000           30.0
9,000,000           45.0
10,000,000           52.5
12,500,000           70.0

(a) What is the cost of equity and cost of capital at this time?
(b) What is the optimal capital structure when bankruptcy costs are considered?
(c) What will the value of the ﬁrm be at this optimal capital structure?

(Optional)   7. Suppose that every investor in the economy is risk-neutral (i.e., does not care about risk), and
that the riskfree rate is 10% (this means that all discounting is done at 10% in this economy).

3
Suppose further that the economy in one year will turn out to be good with a probability of
1/4, medium with a probability of 1/2, and bad with a probability of 1/4.
Now, Xirdneh Imij (XI) is a ﬁrm which has two assets: \$10 million in cash which is invested
in T-bills (and will generate \$11 million for sure in one year), and a one-year project whose
end-of-year cash ﬂow depends on the state of the economy at that time: it will be \$349 million,
\$99 million and \$25 million in the good, medium and bad states respectively.
Assume that XI will cease to exist at the end of the year, i.e., it will return all of its assets at
that time to the appropriate investors. However, if the ﬁrm is not solvent (if its assets do not
generate enough to pay back the bondholders), it will cost \$10 million in legal fees to sort out
who gets what. Also, assume that there are no corporate or personal taxes in this economy.

(a) Suppose that the ﬁrm is currently all-equity ﬁnanced. What is the value of the equity
today?
(b) Now suppose that XI is considering a change in its capital structure. The ﬁrm will raise
\$30 million by issuing one-year bonds which promise an interest rate of 10%. The \$30
million will be used to buy back some equity.
(i) How much does the ﬁrm owe the bondholders in one year (including both interest
and principal)?
(ii) What will the bondholders get in the good, medium and bad state respectively?
(iii) Show that the debt is indeed worth \$30 million today (by discounting the expected
payoﬀ).
(iv) What is the new value of the equity?
(v) Are the shareholders better oﬀ, worse oﬀ, or indiﬀerent? Why?
(c) Suppose instead that XI tries to raise \$40 million by issuing bonds which still promise
an interest rate of 10%.
(i) Show that investors would not purchase the bonds. (Hint: Proceed as in parts
(b)-(i) to (b)-(iii).)
(ii) What is the minimum promised rate (the fair rate) at which investors would start
(iii) Show that, at that fair rate, the shareholders are made worse oﬀ. Why is that the
case?
(d) Suppose that XI decides to go forward with the reﬁnancing in part (b) (i.e., suppose
XI issues \$30 million worth of debt at 10%). Suppose that XI is considering investing
all of its available cash (\$10 million) in a second one-year project that will generate an
end-of-year cash ﬂow of \$23 million, \$13 million and \$3 million in the good, medium and
(i) What is the net present value of the project if taken by itself (i.e., ignoring the ﬁrm’s
other assets and ﬁnancial structure)?
(ii) Show that rational bondholders would have insisted that a clause preventing the
ﬁrm to undertake such a project be added to their debt contract (despite the fact
that the project has a positive NPV).

4
Solutions
1
1. (a) The debt holders will pay tD (rD D) = 3 (15) = 5 in taxes, and so they expect to receive
10 after tax every year.
(b) Before the debt is issued, the ﬁrm pays an expected dividend of EBIT (1 − tc ) =
0.60EBIT. After the debt issue, the ﬁrm’s expected dividend is

(EBIT − rD D)(1 − tc ) = 0.60(EBIT − 15).

The diﬀerence is 0.60 × 15 = 9. That is, because the interest payment is tax-deductible,
the ﬁrm only needs to lower annual dividends by 15(1 − tc ) to aﬀord an annual interest
payment of 15.
(c) The equity-holders used to receive

EBIT (1 − tc )(1 − tE ) = EBIT (1 − 0.40)(1 − 0.15) = 0.51EBIT.

(EBIT − rD D)(1 − tc )(1 − tE ) = (EBIT − 15)(1 − 0.40)(1 − 0.15)
= 0.51(EBIT − 15).

The diﬀerence is 15 × 0.51 = 7.65. That is, their after-tax dividend is reduced by
15(1 − tc )(1 − tE ).
(d) Before the debt issue, the government used to receive the ﬁrm’s earnings that did not
go to equity-holders, namely

EBIT − EBIT (1 − tc )(1 − tE ) = EBIT 1 − (1 − 0.40)(1 − 0.15) = 0.49EBIT.

After the debt issue, the government receives the earnings that do not end up in the
hands of debt-holders and equity-holders, that is,
debt-holders              equity-holders

EBIT − (rD D)(1 − tD ) − (EBIT − rD D)(1 − tc )(1 − tE )
1
= EBIT − 15 1 −          − (EBIT − 15)(1 − 0.40)(1 − 0.15)
3
2
= EBIT − (15) − 0.51(EBIT − 15)
3
2
= 0.49EBIT −     − 0.51 (15)
3
= 0.49EBIT − 2.35.

So the government receives 2.35 less per year.
(e) The eﬀective tax advantage of debt is

(1 − tc )(1 − tE )      (1 − 0.40)(1 − 0.15)
t∗ = 1 −                      = 1−                      = 0.235
1 − tD                     1− 1 3

5
2. Because debt is tax deductible at the corporate level, the eﬀective tax rate is 44% in both
cases.1
(a) If all equity income (including capital gains) is realized immediately, the eﬀective per-
1          1
sonal tax rate on equity income is tE = 2 (0.44) + 2 (0.20) = 32%. Since equity income
is also taxed at the corporate level, a dollar in revenues paid via the equity channel
becomes
(1 − tc )(1 − tE ) = (1 − 0.35)(1 − 0.32) = 0.442
in the hands of equity-holders. The eﬀective tax rate is therefore 1 − 0.442 = 55.8%.
(b) If only dividends are taxed when paid and capital gains can be deferred forever, the
1          1
eﬀective personal tax rate on equity income is tE = 2 (0.44) + 2 (0) = 22%. Since equity
income is also taxed at the corporate level, a dollar in revenues paid via the equity
channel becomes
(1 − tc )(1 − tE ) = (1 − 0.35)(1 − 0.22) = 0.507
in the hands of equity-holders. The eﬀective tax rate is therefore 1 − 0.507 = 49.3%.
3. (a) Suppose at ﬁrst that D = 0 and tc = tE = tD = 0. Since the ﬁrm is all-equity ﬁnanced,
we have rA = rE = 0.20. The expected earnings before interest and taxes are

EBIT = (0.1)(1,000) + (0.4)(2,000) + (0.5)(4,200) = 3,000.
Therefore:
(i)   VU = EU = EBIT = 3,000 = 15,000;
rA       0.20
(ii)   P = EU /NU = 15,000/1,000 = 15;
3,000
(iii)   EPS = EBIT = 1,000 = 3;
NU
(iv)    rA = rE = 0.20.
(b) Now assume that the ﬁrm issues \$7,500 of debt at rD = 0.10 and uses the proceeds to
repurchase shares.
Since we are assuming no taxes, we have VL = VU = 15,000;
(i)
DL = 7,500;
(ii)
EL = VL − DL = 7,500;
(iii)
(iv)The total wealth of the shareholders is 15,000, as they still have EL and they receive
\$7,500 in cash from the debt issue. On a per-share basis, this amounts to P =
15,000                                                               7,500
1,000 = 15. The number of shares repurchased is therefore n = 15 = 500, and
there are now 1,000 − 500 = 500 shares outstanding.2
(v) The required (expected) return on equity can be calculated using what the equity-
holders receive every year (earnings after interest payment) and the value of their
equity (EL ):
EBIT − rD DL   3,000 − 750   2,250
rE =                =             =       = 0.30.
EL            7,500      7,500
1
Note that this is because the debt is assumed to be perpetual, and so it is paid in the form of interest forever.
2
Note that we could have solved this more analytically. We have (1,000−n)P = EL = 7,500 and nP = DL = 7,500.
We can then solve for n = 500 and P = 15.

6
Alternatively, we could have calculated rE using the levering formula:3
DL
rE = rA +         (1 − tc )(rA − rD )
EL
7,500
= 0.20 +          (1 − 0)(0.20 − 0.10) = 0.30.
7,500
(vi) Without taxes, WACC should be equal to rA and rE . Let us verify this:
DL            EL
WACCL =            rD +          rE = 0.5(0.1) + 0.5(0.3) = 0.20.
VL            VL
Notice that the value of the ﬁrm is unaﬀected by the increase in debt. This is because
the reshuﬄing of claims to the cash ﬂows generated by the ﬁrm’s assets has not changed
(a) the cash ﬂows generated by the ﬁrm’s assets (still equal to \$3,000 now split up as
dividends of \$2,250 to shareholders and interest of \$750 to debt-holders) nor (b) the
riskiness of those cash ﬂows. Hence, the discount rate applicable has not changed and
the ﬁrm’s weighted average cost of capital remains unchanged. The required return on
common stock is now greater than the expected return on assets (i.e., rE > rA ): because
the bondholders are paid ahead of the stockholders, the stockholders face more risk and
so demand compensation for bearing that risk.
(c) (i) The value of the unlevered ﬁrm will change when we introduce corporate taxes, as
the government now receives a fraction of the cash ﬂows generated by the ﬁrm’s
assets. The after-tax cash ﬂows generated by the assets of the ﬁrm will now be
(1 − tc )EBIT = (1 − 0.4)3,000 = 1,800.
Thus, the value of the unlevered ﬁrm is
(1 − tc )EBIT   (1 − tc )EBIT   1,800
VU =                 =               =       = 9,000.
rA              rE         0.20
The value of the levered ﬁrm is then
VL = VU + tc DL = 9,000 + 0.4(7,500) = 12,000.
(ii) The value of debt remains at DL = 7,500: the debt-holders pay \$7,500 and receive
a claim worth \$7,500.
(iii) The value of the equity is now
EL = VL − DL = 12,000 − 7,500 = 4,500.
(iv) Upon the announcement of the debt issue (but before the debt is actually issued),
the value of the ﬁrm goes up to VL = VU +tc DL = 9,000+0.4(7,500) = 12,000. Since
there are still 1,000 shares outstanding at this point, each share must be trading at
P = 12,000 = 12. The proceeds from the debt issue can therefore be used to buy
1,000
back n = 7,500 = 625 shares.4
12
3
We know that rA is unaﬀected by the capital structure, as the ﬁrm’s projects have not changed.
4
Again, we could have solved this in a more analytical manner. We have (1,000 − n)P = EL = 4,500 and
nP = DL = 7,500. We can then solve for n = 625 and P = 12.

7
(v) The required rate of return on equity is again

(1 − tc )(EBIT − rD DL )   (1 − 0.4)(3,000 − 750)
rE =                             =                        = 0.30.
EL                       4,500

(vi) However, the weighted average cost of capital for the ﬁrm is now

(1 − tc )EBIT   (1 − 0.4)3,000
WACCL =                  =                = 0.15.
VL            12,000
Notice that alternatively

DL                 EL
WACCL = (1 − tc )rD                  + rE
VL                 VL
7,500                    4,500
= (1 − 0.4)(0.10)                     + (0.3)             = 0.15.
12,000                  12,000

Bankruptcy risks would reduce the amount by which the ﬁrm’s value would increase due
to leverage and would increase the weighted average cost of capital.
(d) If tc > 0, tD > 0 and tE = 0, then
t∗

1 − tc
VL = VU + DL 1 −                ,
1 − tD

where we still have VU = 9,000 since rE = rA and so
(1 − tE )(1 − tc )EBIT   (1 − 0)(1 − 0.4)3,000
VU =                            =                       = 9,000.
(1 − tE )rE             (1 − 0)0.20

Also, DL = 7,500 in all three cases (the bondholders get what they pay for). Therefore,
if
(i) tD = 0.3, then

1 − 0.4
VL = 9,000 + 7,500 1 −         = 10,071.43, and
1 − 0.3
EL = VL − DL = 10,071.43 − 7,500 = 2,571.43;

(ii) tD = 0.4, then

1 − 0.4
VL = 9,000 + 7,500 1 −         = 9,000,              and
1 − 0.4
EL = VL − DL = 9,000 − 7,500 = 1,500;

4. (a) The initial value (in \$million) of Gladstone’s equity is

(0.25)(150) + (0.25)(135) + (0.25)(95) + (0.25)(80)
E=                                                       = 109.52.
1.05

8
(b) Gladstone will be able to repay its promise (of \$100 million) to debt-holders in the ﬁrst
two states, but not in the last two (it will pay whatever is available instead). The value
of the debt is therefore
(0.25)(100) + (0.25)(100) + (0.25)(95) + (0.25)(80)
D=                                                        = 89.29.
1.05
(c) For a zero-coupon bond, the yield to maturity is the promised interest rate on the debt
contract. In this case, it is
100 − 89.29
y=                = 12.0%.
89.29
The expected rate of return is the rate that debt-holders can expect from this debt. By
deﬁnition, it must be equal to rD , which is equal to rf (i.e., 5%) in this problem. Still, let
us verify it. The debt-holders expect to receive (0.25)(100) + (0.25)(100) + (0.25)(95) +
(0.25)(80) = 93.75. Since their debt is currently worth 89.29, their expected rate of
return is
93.75 − 89.29
rD =                  = 5.0%.
89.29
(d) Gladstone’s equity-holders receive the value that is available after the debt-holders are
paid. Thus, the initial value of Gladstone’s equity is
(0.25)(50) + (0.25)(35) + (0.25)(0) + (0.25)(0)
E=                                                    = 20.24.
1.05
The total value of the ﬁrm is still5

V = D + E = 89.29 + 20.24 = 109.52.

5. (a) The initial value (in \$million) of Gladstone’s equity is as before:

(0.25)(150) + (0.25)(135) + (0.25)(95) + (0.25)(80)
E=                                                        = 109.52.
1.05
(b) As before, Gladstone will be able to repay its promise (of \$100 million) to debt-holders
in the ﬁrst two states, but not in the last two (i.e., in the two bad states). In the two
bad states, 25% of the value will be lost, as the ﬁrm goes into bankruptcy. Thus, there
will only be 95(1 − 0.25) = 71.25 and 80(1 − 0.25) = 60 to repay the debt-holders in
these two states. The value of the debt is therefore
(0.25)(100) + (0.25)(100) + (0.25)(71.25) + (0.25)(60)
D=                                                           = 78.87.
1.05
(c) For a zero-coupon bond, the yield to maturity is the promised interest rate on the debt
contract. In this case, it is
100 − 78.87
y=                = 26.8%.
78.87
5
In my calculations, kept all the decimals. This is why there is a small rounding error.

9
The expected rate of return is the rate that debt-holders can expect from this debt.
By deﬁnition, it must be equal to rD , which is equal to rf (i.e., 5%) in this problem.
Still, let us verify it. The debt-holders expect to receive (0.25)(100) + (0.25)(100) +
(0.25)(71.25) + (0.25)(60) = 82.81. Since their debt is currently worth 78.87, their
expected rate of return is
82.81 − 78.87
rD =                 = 5.0%.
78.87
(d) Gladstone’s equity-holders receive the value that is available after the debt-holders are
paid. Thus, the initial value of Gladstone’s equity is
(0.25)(50) + (0.25)(35) + (0.25)(0) + (0.25)(0)
E=                                                   = 20.24.
1.05
The total value of the ﬁrm is now

V = D + E = 78.87 + 20.24 = 99.11.

This is smaller than in part (a), when the ﬁrm was unlevered. The diﬀerence is due to
the present value of bankruptcy costs which is
(0.25)(0) + (0.25)(0) + (0.25)(0.25 × 95) + (0.25)(0.25 × 80)
P V (bankruptcy costs) =
1.05
= 10.42.

\$109.52
(e) Using the value of the ﬁrm calculated in part (a), the price per share is     10      = \$10.95.
(f) As soon as the Gladstone announces its plan to issue the debt and repurchase shares,
the value of the ﬁrm will drop to \$99.11, as calculated in part (d). This means that
the share price will drop to \$99.11 = 9.91. This also means that the debt issued by
10
Gladstone will allow it to raise \$78.87 (from part (b)) and to repurchase \$78.87 = 7.96
\$9.91
million shares. The other 2.04 million shares also trade at \$9.91 per share, and so are
worth \$20.24 (which we found in part (d)).

6. (a) The cost of equity is given by the CAPM:

rE = rf + (rm − rf )βE = 0.09 + (0.06)1 = 15%.

Since the ﬁrm is all equity ﬁnanced, the cost of capital is equal to the cost of equity.
Notice that this implies that Nadir’s earnings are expected to grow at 5% per year as
EBIT (1 − tc )   2,000,000(1 − 0.40)
V =                  =                        ⇒       g = 5%.
rE − g               0.15 − g

(b) We from the lecture notes that

VL = VU + P V (tax shield) − P V (bankruptcy costs),

so that we should try to ﬁnd the debt level which maximizes

P V (tax shield) − P V (bankruptcy costs).

10
The following table shows the present value of the debt tax shield and the present value
of bankruptcy costs and their diﬀerence for the diﬀerent debt levels.

Value of        Probability of       P V (tax shield)       P V (bank. costs)       Value Added
Debt [D]         Failure [pD ]          [0.4 × D]              [pD × 8M]              of Debt
2,500,000             0.000              1,000,000                      0             1,000,000
5,000,000             0.080              2,000,000                640,000             1,360,000
7,500,000             0.205              3,000,000              1,640,000             1,360,000
8,000,000             0.300              3,200,000              2,400,000               800,000
9,000,000             0.450              3,600,000              3,600,000                     0
10,000,000             0.525              4,000,000              4,200,000              -200,000
12,500,000             0.700              5,000,000              5,600,000              -600,000

Obviously, the optimal debt level is \$5,000,000 or \$7,500,000.6
(c) With this optimal capital structure, the total value of the ﬁrm is

VL = VU + value added of debt = 12,000,000 + 1,360,000 = 13,360,000.

7. (a) XI’s end-of-year assets in the diﬀerent states of the economy will be:7

State of the economy
(1/4)     (1/2)     (1/4)
Cash                   11         11          11
Project                349        99          25
Total Assets           360       110          36

If the ﬁrm is all-equity ﬁnanced, the equity is worth
360(1/4) + 110(1/2) + 36(1/4)
EU = VU =                                    = 140.
1.10
(b) (i) The ﬁrm will owe \$30 million plus 10% interest, i.e. \$33 million.
(ii) The ﬁrm’s assets at the end of the year will be suﬃcient to pay the \$33 million that
it owes to the bondholders in all three states of the economy.
(iii) The debt is worth
33(1/4) + 33(1/2) + 33(1/4)
DL =                                = 30.
1.10
(iv) The equity is now worth

(360 − 33)(1/4) + (110 − 33)(1/2) + (36 − 33)(1/4)
EL =                                                       = 110.
1.10
6
More precisely, since we are not given the probabilities of failure for all the debt levels between \$5,000,000 and
\$7,500,000, we can only conclude that the optimal debt level is somewhere between \$5,000,000 and \$7,500,000.
7
All the ﬁgures in this solution are in \$million.

11
(v) The shareholders got \$30 million in cash (from the share repurchase) and still have
\$110 million in equity. Therefore, their total wealth is still \$140 million, which
(c) (i) The bonds now promise \$44 million, which implies that the ﬁrm will not be solvent
in the bad state of the economy (since the ﬁrm only has \$36 million worth of assets
in that state). As a result, bankruptcy costs of \$10 million will be incurred in
the bad state, and so the bondholders will only receive \$36 million − \$10 million =
\$26 million in that state. Their debt is then worth
44(1/4) + 44(1/2) + 26(1/4)
DL =                               = 35.909,
1.10
which is less than what they pay for it. Obviously, investors would simply refuse to
(ii) We would like to ﬁnd the promised amount A, which will make the debt worth
\$40 million:
A(1/4) + A(1/2) + 26(1/4)
40 =                                ⇒ A = 50.
1.10
So the ﬁrm has to promise the bondholders 50 − 1 = 25% in order for them to expect
40
a return of 10% on average (and therefore accept to buy the bonds).
(iii) The shareholders will then receive nothing in the bad state of the economy. Their
equity is now worth

(360 − 50)(1/4) + (110 − 50)(1/2) + (0)(1/4)
EL =                                                = 97.727.
1.10
Since they sold back some equity for \$40 million, this means that their total wealth is
now \$97.727 million+\$40 million = \$137.727 million, which is less than the \$140 mil-
lion that they originally had. The reason why their wealth is now lower is that the
bondholders get what they pay for (they are not fooled by the potential bankruptcy
costs). As a result, the bankruptcy costs are imposed on the shareholders, who lose

10(1/4)
P V (bankruptcy costs) =           = 2.273
1.10
in wealth.
(d) (i) The present value P V of the expected end-of-year cash ﬂow of the project is

(23)(1/4) + (13)(1/2) + (3)(1/4)
PV =                                    = 11.818.
1.10
Since the project require an initial investment of \$10 million, its net present value
is therefore
NP V = −10 + 11.818 = 1.818 > 0.

12
(ii) If XI undertakes this second project, the ﬁrm’s end-of-year assets in the three dif-
ferent states will be as follows:

State of the economy
(1/4)     (1/2)     (1/4)
1st project         349       99        25
2nd project         23        13         3
Total Assets        372      112        28

This means that the ﬁrm would no longer be solvent in the bad state (the \$28 million
in assets is not suﬃcient to pay the \$33 million that was promised to the bondhold-
ers), so that

33(1/4) + 33(1/2) + (28 − 10)(1/4)
DL =                                      = 26.591 < 30.
1.10

13

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 0 posted: 2/16/2013 language: Unknown pages: 13
How are you planning on using Docstoc?