Solutions
to Practice
Problems – Chapters 18,
19
1 a) NPV of project =
(.3*2,200 + .4 * 300 + .3 * 500)/1.04 – 1,000 = -105.77
b) If the project is not
taken, the $1,000 cash will be invested for one period at 4%, generating a
cashflow of $1,040 for sure at time 1. Cashflows
will then be as follows:
State of Economy Probability
Total CF CF
to b/h CF to s/h
Great
0.3
11,940
10,000
1,940
Okay
0.4
7,940
7,940
0
Poor
0.3
5,940
5,940
0
The value of the firm is (.3*11,940
+ .4*7,940 + .3*5,940)/1.04 = 8,211.54
The value of debt is (.3*10,000 +
.4*7,940 + .3*5,940)/1.04 = 7,651.92
The value of equity is (.3*1,940 +
.4*0 + .3*0)/1.04 = 559.62
c) If the project is taken,
cashflows are as follows:
State of Economy Probability
Total CF CF
to b/h CF to s/h
Great
0.3
13,100
10,000
3,100
Okay
0.4
7,200
7,200
0
Poor
0.3
5,400
5,400
0
The value of the firm is (.3*13,100
+ .4*7,200 + .3*5,400)/1.04 = 8,105.77
The value of debt is (.3*10,000 +
.4*7,200 + .3*5,400)/1.04 = 7,211.54
The value of equity is (.3*3,100 +
.4*0 + .3*0)/1.04 = 894.23
Stockholders will invest in this
project, even though it is a negative NPV project, because it increases their
wealth.
2 a) NPV of project =
(.2*1,800 + .5 * 500 + .3 * 300)/1.05 – 1,000 = -105.77
b) If the project is not
taken, the $750 cash will be invested for one period at 5%, generating a
cashflow of $787.50 for sure at time 1. Cashflows
will then be as follows:
State of Economy Probability
Total CF CF
to b/h
CF to s/h
Great
0.2
6,987.50
5,000
1,987.50
Okay
0.5
3,187.50
3,187.50
0
Poor
0.3
1,987.50
1,987.50 0
The value of the firm is
(.2*6,987.50 + .5*3,187.50 + .3*1,987.50)/1.05 = 3,416.67
The value of debt is (.2*5,000 +
.5*3,187.50 + .3*1,987.50)/1.05 = 3,038.10
The value of equity is (.2*1,987.50
+ .5*0 + .3*0)/1.04 = 378.57
c) If the project is taken,
cashflows are as follows:
State of Economy Probability
Total CF CF
to b/h CF to s/h
Great
0.2
8,000
5,000
3,000
Okay
0.5
2,900
2,900
0
Poor
0.3
1,500
1,500
0
The value of the firm is (.2*8,000
+ .5*2,900 + .3*1,500)/1.05 = 3,333.33
The value of debt is (.2*5,000 +
.5*2,900 + .3*1,500)/1.05 = 2,761.90
The value of equity is (.2*3,100 +
.5*0 + .3*0)/1.05 = 571.43
Stockholders will invest in this
project, even though it is a negative NPV project, because it increases their
wealth.
3 a) NPV of project =
(.3*900 + .45 * 1,300 + .25 * 1,500)/1.04 – 1,000 = 182.69
b) If the project is not
taken, cashflows will be as follows:
State of Economy Probability
Total CF
CF to b/h
CF to s/h
Great
0.3
12,000
9,000
3,000
Okay
0.45 8,500
8,500
0
Poor
0.25
6,000
6,000
0
The value of the firm is (.3*12,000
+ .45*8,500 + .25*6,000)/1.04 = 8,581.73
The value of debt is (.3*9,000 +
.45*8,500 + .25*6,000)/1.04 = 7,716.35
The value of equity is (.3*3,000 +
.45*0 + .25*0)/1.04 = 865.38
c) If the project is taken,
cashflows are as follows:
State of Economy Probability
Total CF CF
to b/h CF
to s/h
Great
0.3
12,900
9,000
3,900
Okay
0.45
9,800
9,000
800
Poor
0.25
7,500
7,500
0
The value of the firm is (.3*12,900
+ .45*9,800 + .25*7,500)/1.04 = 9,764.42
The value of debt is (.3*9,000 +
.45*9,000 + .25*7,500)/1.04 = 8,293.27
The value of equity is (.3*3,900 +
.45*800 + .25*0)/1.04 = 1,471.15
d) Stockholders initially had
$1,000 in cash plus $865.38 in equity, for total wealth of $1,865.38.
By taking the project, this drops to $1,471.15.
They will not invest in this project, even though it is a positive NPV
project, because it decreases their wealth.
4 a) NPV of project =
(.4*750 + .35 * 1,000 + .25 * 1,250)/1.05 – 800 = 116.67
b) If the project is not
taken, cashflows will be as follows:
State of Economy Probability
Total CF CF
to b/h CF
to s/h
Great
0.4
8,500
6,500
2,000
Okay
0.35
7,000
6,500
500
Poor
0.25
5,500
5,500
0
The value of the firm is (.4*8,500
+ .35*7,000 + .25*5,500)/1.05 = 6,880.95
The value of debt is (.4*6,500 +
.35*6,500 + .25*5,500)/1.05 = 5,952.38
The value of equity is (.4*2,000 +
.35*500 + .25*0)/1.05 = 928.57
c) If the project is taken,
cashflows are as follows:
State of Economy Probability
Total CF
CF to b/h
CF to s/h
Great
0.4
9,250
6,500
2,750
Okay
0.35
8,000
6,500
1,500
Poor
0.25
6,750
6,500
250
The value of the firm is (.4*9,250
+ .35*8,000 + .25*6,750)/1.05 = 7,797.62
The value of debt is (.4*6,500 +
.35*6,500 + .25*6,500)/1.05 = 6,190.48
The value of equity is (.4*2,750 +
.35*1,500 + .25*250)/1.05 = 1,607.14
d) Stockholders initially had $800
in cash plus $928.57 in equity, for total wealth of $1,728.57.
By taking the project, this drops to $1,607.14.
They will not invest in this project, even though it is a positive NPV
project, because it decreases their wealth.
5 a) With just corporate taxes, we have
WACC = (D/V) Rd (1-Tc) + (E/V) Re =>
0.13 = 0.15*Rd*.65
+ 0.85*0.145
=> Rd
= (.13 - .85*.145)/(.15*.65) = 6.9231%
The cost of capital if the firm was unlevered is Ra. We have
WACC = Ra * [ 1 – (D/V)* Tc] =>
.13 = Ra*(1 - .15*.35)
=> Ra
= .13/(1 - .15*.35) = 13.67203%
b) With corporate and personal taxes, we have
T’ = 1
– [(1 - Tpe)(1 - Tc)]/( 1 - Tp) = 1 – (1 -
.2)*(1 - .35)/(1 – 0.33) = .2239
WACC = (D/V) Rd (1-T’) + (E/V) Re =>
0.13 = 0.15*Rd*(1
- 0.2239) + 0.85*0.145
=> Rd
= (.13 - .85*.145)/(.15*.7761) = 5.7981%
Also, WACC = Ra * [ 1 – (D/V)* T’] =>
.13 = Ra*(1 - .15*.2239)
=> Ra
= .13/(1 - .15*.2239) = 13.4517%
6 a) D/E = .9
=> D = .9*E
=> D/V =
.9E/(E+.9E) = .9/1.9 = .4737
WACC = (D/V) Rd (1-T’) + (E/V) Re = .4737*.12 * (1 - .2) + (1 - .4737)*.156 = 12.7579%
b) Unlike PCM, now WACC changes as you change leverage.
If D/E falls to 2/3:
D/V = (2/3)/(1 + 2/3) = .4
At the new debt level, both Rd and Re will change, so we can’t use the weighted average formula.
Since Ra stays the same, we need to first compute Ra and then use that to get WACC.
At the original debt level:
WACC = Ra * [ 1 – (D/V)* T’] => .127579
= Ra * [ 1 – .4737 * .2]
=> Ra = .12757/[ 1 – .4737 * .2] = 14.093%
At the new debt level:
WACC = Ra * [ 1 – (D/V)* T’] = .14093 * (1 – .4*.2) = 12.9656%
7 a)
0.12 = 0.45*Rd*(1 - .2) + 0.55*0.18
=> Rd
= (.12 - .55*.18)/(.45*.8) = 5.8333%
b) The new cost of debt would be 1.2*.058333 = 7%
Once again we can’t use the weighted average formula
for WACC, since Re also changes.
Once again, we first compute Ra.
WACC = Ra * [ 1 – (D/V)* T’] => .12
= Ra * [ 1 – .45 * .2]
=> Ra = .12/[ 1 – .45 * .2] = 13.1868%
Then, at the new debt level, WACC = Ra * [ 1 – (D/V)* T’] = .131868 * (1 – .6*.2) = 11.6044%
Re = Ra + (Ra – Rd)*(D/E)
= .116044 + (.116044 - .07)*(.6/.4) = 20.611%
8 a) Since we don’t know the debt ratio, we’ll have
to use the APV approach.
VU = 220,000/0.11 = 2,000,000
D = 800,000 (given)
PVTS = D*Tc = 800,000*.35 = 280,000
VL = 2,000,000 + 280,000 = 2,280,000
E = VL - D = 2,280,000 - 800,000 = 1,480,000
Re = Ra + (Ra – Rd)*(D/E)*(1
- Tc) = .11 + (.11 - .06)*(800/1480)*0.65 = 12.7568%
WACC = Ra * [ 1 – (D/V)* Tc] = .11 * [1 – (800/2280)*.35] = 9.6491%
b) Once again, T’ = .2239 (as in problem 1). We
repeat the computations using T’ in place of Tc.
PVTS = D*T’ = 800,000*.2239 = 179,104.48
VL = 2,000,000 + 179,104.480 = 2, 179,104.48
E = VL - D = 2, 179,104.48 - 800,000 =
1,379,104.48
Re = Ra + (Ra – Rd)*(D/E)*(1
- T’) = .11 +
(.11 - .06)*(800/1379.10)*(1 - .2239) = 13.2511%
WACC = Ra * [ 1 – (D/V)* T’] = .11 * [1 – (800/2179.1)*.2239] = 10.0959%
9) WACC = (D/V) Rd*(1 - T’) + (E/V) Re = (8/25)*.115*.8 + (17/25)*.165 = 14.164%
Ra
= WACC/[ 1 – (D/V)* T’] = .14164/[1 – (8/25)*.2] =
15.1325%
10 a) VU = 1,250,000/.09625 = 12,987,012.99
D = 300,000/.06 = 5,000,000
PVTS = 5,000,000 * .2 = 1,000,000
VL = 12,987,012.99 + 1,000,000 = 13,987,012.99
E = 13,987,012.99 - 5,000,000 = 8,987,012.99
b) Re = Ra + (Ra – Rd)*(D/E)*(1
- T’) = .09625 +
(.09625 - .06)*(5/8.987)*.8 = 11.2384%
c) WACC = Ra * [ 1 – (D/V)* T’] = .09625 * [1 – (5/13.987)*.2] = 8.9369%
11) WACC = Ra * [ 1 – (D/V)* T’] = .095 * [1 – 0.25*.2] = 9.025%
VL = 1,500,000/.09025 = 16,620,498.61
VU = 1,500,000/.095 = 15,789,473.68
Increase in the value of the firm due to leverage =
16,620,498.61 - 15,789,473.68 = 831.024.93
12 a) To compute WACC, we’ll first need to get VL
using the APV method.
D = 120,000/.0725 = 1,655,172.41
PVTS = 1,655,172.41 * .2 = 331,034.48
VU = 575,000/.1 = 5,750,000
VL = 5,750,000 + 331,034.48 = 6,081,034.48
WACC = Ra * [ 1 – (D/V)* T’] = .1 * [1 – (1.655/6.081)*.2] = 9.4556%
(Note that we can also go WACC = CF/ VL =
575,000/6,081,034.48 = 9.4556%)
b) E = 6,081,034.48 - 1,655,172.41 = 4,425,862.07
c) At the new debt ratio, WACC = Ra*[ 1 – (D/V)* T’] = .1 * [1 – .4*.2] = 9.2%
VL = 575,000/.092 = 6,250,000
E = .6*6,250,000 = 3,750,000
13) Estimating WACC still
follows the same procedure as in Chapter 9.
All that changes is the formulas we use to compute Ra
from Re, and then WACC from Ra. First
we compute Ra for each
firm, then use the average Ra
as the estimated Ra for
the project. Combining the average Ra
with the project’s D/V ratio gives the project’s WACC.
Firm A:
WACC
= (D/V) Rd*(1
- T’)
+ (E/V) Re
= (115/555)*.0599*.8 + (440/555)*.124 = 10.8236%
Ra
= WACC/[ 1 – (D/V)* T’] = .108236/[1 – (115/555)*.2]
= 11.2915%
Firm B:
WACC
= (155/715)*.0602*.8 + (560/715)*.128 = 11.0692%
Ra
= .110692/[1 – (155/715)*.2] = 11.5709%
Firm C:
WACC
= (184/864)*.06*.8 + (680/864)*.126 =
10.9389%
Ra
= .109389/[1 – (184/864)*.2] = 11.4255%
Firm D:
WACC
= (220/990)*.0605*.8 + (770/990)*.132 =
11.3422%
Ra
= .113422/[1 – (220/990)*.2] = 11.8698%
The project’s Ra is then the simple average of the Ra for these four firms, which is 11.5394%.
The
project’s WACC = .115394 * (1 - .4 * .2) = 10.6163%.
14 a) Firm A:
WACC = Ra * [ 1 – (D/V)* T’] = .131 * [1 – .33*.2] = 12.2354%
Re = Ra + (Ra – Rd)*(D/E)*(1
- T’) = .131 +
(.131 - .091)*(.33/.67)*.8) = 14.6761%
Firm B:
WACC
= (D/V) Rd*(1
- T’)
+ (E/V) Re
= .36*.094*.8 + .64*.1489 = 12.2368%
Ra
= .122368/[1 – .36*.2] = 13.1862%
Firm C:
Ra
= .1206/[1 – .42*.2] = 13.1659%
Re = Ra + (Ra – Rd)*(D/E)*(1
- T’) = .131659 +
(.131659 - .096)*(.42/.58)*.8) = 15.2317%
Firm D:
WACC = (D/V)
Rd*(1 - T’)
+ (E/V) Re => .121968
= .38* Rd *.8 + .62*.1458
=>
Rd = (.121968 -
.62*.1458)/(.38*.8) = 10.3855%
b)
Once again, the project’s Ra
is the simple average of the Ra for
these four firms = (.1310 + .131862 + .131659 + .132)/4 = 13.163%
The project’s WACC = .13163 * (1 - .35*.2) = 12.2416%
15 a) Base case NPV =
15,900/.14 – 111,000 = 2,571.43
Amount of new equity to be
issued = 111,000*.5/(1 - .055) = 58,730.16
Issue costs for equity =
.055*58,730.16 = 3,230.16
Amount of new debt to be
issued = 111,000*.5/(1 - .025) = 56,923.08
Issue costs for debt =
.025*56,923.08 = 1,423.08
Total issue costs = 3,230.16
+ 1,423.08 = 4,653.24
PVTS = D*T’
= 56,923.08 * .2 = 11,384.62
Adjusted NPV of project =
Base case NPV + PVTS - Issue costs = 2,571.43 + 11,384.62 - 4,653.24 = 9,302.81
b) For computing WACC we use
the value of the project before issue costs. In other words, V = NPV + investment + Issue costs = 9,302.81
+ 111,000 + 4,653.24 = 124,956.04
WACC = .14 * [1 –
(56,923.08/124,956.04)*.2] = 12.7245%
16 a) Base case NPV =
305,000/.12 – 2,500,000 = 41,666.67
Amount of new equity to be
issued = 2,500,000*.6/(1 - .06) = 1,595,744.68
Issue costs for equity =
.06*1,595,744.68 = 95,744.68
Amount of new debt to be
issued = 2,500,000*.4/(1 - .04) = 1,041,666.67
Issue costs for debt =
.04*1,041,666.67 = 41,666.67
Total issue costs =
95,744.68 + 41,666.67 = 137,411.35
PVTS = D*T’
= 1,041,666.67 * .2 = 208,333.33
Adjusted NPV of project =
Base case NPV + PVTS - Issue costs = 41,666.67 + 208,333.33 - 137,411.35 =
112,588.65
b) For computing WACC we use
the value of the project before issue costs. In other words, V = NPV + investment + Issue costs =
112,588.65 + 2,500,000 + 137,411.34 = 2,750,000
WACC = .12 * [1 –
(1,041,666.67/1,041,666.67)*.2] = 11.0909%
17 a) Base case NPV =
110,000/.13 – 1,000,000 = -120,000
Amount of new equity to be
issued = 1,000,000*.75/(1 - .05) = 789,473.68
Issue costs for equity =
.05*789,473.68 = 39,473.68
Amount of new debt to be
issued = 1,000,000*.25/(1 - .03) = 257,731.96
Issue costs for debt =
.03*257,731.96 = 7,731.96
Total issue costs =
39,473.68 + 7,731.96 = 47,205.64
PVTS = D*T’
= 257,731.96 * .2 = 51,546.39
Adjusted NPV of project =
Base case NPV + PVTS - Issue costs = -120,000 + 51,546.39 - 47,205.64 =
-115,659.25
b) For computing WACC we use
the value of the project before issue costs. In other words, V = NPV + investment + Issue costs =
-115,659.25 + 1,000,000 + 47,205.64 = 931,546.39
WACC = .13* [1 –
(257,731.96/931,546.39)*.2] = 11.8083%
18 a) Base case NPV =
70,000/.12 – 500,000 = 83,333.33
Amount of new equity to be
issued = 500,000*.8/(1 - .05) = 421,052.63
Issue costs for equity =
.05*421,052.63 = 21,052.63
Amount of new debt to be
issued = 500,000*.2/(1 - .025) = 102,564.10
Issue costs for debt =
.025*102,564.10 = 2,564.10
Total issue costs =
21,052.63 + 2,564.108 = 23,616.73
PVTS = D*T’
= 102,564.10 * .2 = 20,512.82
Adjusted NPV of project =
Base case NPV + PVTS - Issue costs = 83,333.33 + 20,512.82 - 23,616.73 =
80,229.42
b) For computing WACC we use
the value of the project before issue costs. In other words, V = NPV + investment + Issue costs =
80,229.42 + 500,000 + 23,616.73 = 603,846.15
WACC = .12 * [1 –
(102,564.10/603,846.15)*.2] = 11.5924%
19 a)
If there were no bankruptcy costs, the firm’s capital structure decision would
not be affected by the probability of bankruptcy. T If there are no
bankruptcy costs, the combined cashflows to stockholders and bondholders are not
affected; so the combined value of debt and equity (i.e. VL) is not
affected.
b)
If there were no bankruptcy costs or agency costs of debt in the real
world, the marginal benefit of issuing debt would remain constant at T’.
F The marginal benefit would be constant
initially, but then it would start to decline, and drop all the way to zero, due
to non-utilization of tax shields.
c)
A levered firm in financial distress may sometimes accept a negative NPV
project if the cashflows from the project are negatively correlated with
existing cashflows. F A negative NPV
project is accepted if the cashflows from the project are positively correlated
with existing cashflows. It’s
only then that stockholders get the benefit if the project makes a big payoff.
d)
When the stockholders of a levered firm end up accepting a negative NPV
project or rejecting a positive NPV project, their wealth increases by doing so;
hence, this represents a benefit of issuing debt.
F At the time they issue debt, the
market realizes they will accept some negative NPV projects and reject some
positive NPV project in the future; this reduces PVGO and therefore VL.
Since stockholder wealth is reduced, it is a cost of issuing debt.
e)
When a levered firm under-invests (rejects a positive NPV project), the
reason for this is that stockholders have to put up the entire investment, but
too much of the resulting cashflows go to bondholders. T
The project’s cashflows are negatively correlated with the existing
assets. It makes a big payoff when
the firm was otherwise going to default. A
significant part of the project’s cashflows goes to the bond-holders.
The cashflows that end up going to stockholders are insufficient to
justify the investment.
f)
Different firms have different marginal costs and marginal benefits of
issuing debt, leading them to choose a different debt ratio.
T Marginal benefit starts at the
same level for everyone (T’) but the rate at which it declines
depends on the likelihood of not being able to fully utilize interest tax
shields. Similarly, marginal
cost increases faster for firms with a higher probability of bankruptcy, or
firms that stand to lose a larger proportion of asset value if they go bankrupt.
g)
A more capital intensive firm will tend to have a higher debt ratio.
F Being more capital intensive
means that depreciation tax shields are large; this make sit less likely that
interest tax shields will be fully utilized, and
marginal benefit declines faster, leading to a lower debt ratio.
h)
When we use either the APV method or the WACC method to compute a
project’s NPV, we are taking the effect of debt financing into account instead
of ignoring it. Hence, interest
charges should not be ignored in computing a project’s net cashflows. F If interest charges
are included while computing cashflows, in addition to being taken into account
when computing NPV, that would be double counting.
i) If
interest charges are deducted when computing net cashflows, that gives the
cashflows to stockholders, instead of the total cashflows generated by the
assets. F To get the
cashflows to stockholders, we also have to add the annual interest tax shields.
j)
Managers who know their shares are under-priced prefer to issue debt
rather than equity; managers who know their shares are over-priced prefer to
issue equity rather than debt. F
Both types of managers prefer to issue debt rather than equity.
The managers who know their shares are over-priced do so to avoid
revealing to the market that they are underpriced.
k) Internally
generated funds represent the preferred source of capital for all firms, since
there are neither any issue costs nor any inside information problems associated
with them. T