4 Chapter problems
1. We need to find the annuity payment in retirement. Our retirement savings ends at the
same time the retirement withdrawals begin, so the PV of the retirement withdrawals
will be the FV of the retirement savings. So, we find the FV of the stock account and the FV
of the bond account and add the two FVs.
Stock account: FVA = $700[{[1 + (.10/12) ]360 1} / (.10/12)] = $1,582,341.55
Bond account: FVA = $300[{[1 + (.06/12) ]360 1} / (.06/12)] = $301,354.51
So, the total amount saved at retirement is:
$1,582,341.55 + 301,354.51 = $1,883,696.06
Solving for the withdrawal amount in retirement using the PVA equation gives us:
PVA = $1,883,696.06 = C[1 {1 / [1 + (.08/12)]300} / (.08/12)]
C = $1,883,696.06 / 129.5645 = $14,538.67 withdrawal per month
2. Here, we are trying to find the interest rate when we know the PV and FV. Using the FV
equation:
FV = PV(1 + r)
$4 = $3(1 + r)
r = 4/3 1 = 33.33% per week
The interest rate is 33.33% per week. To find the APR, we multiply this rate by the
number of weeks in a year, so:
APR = (52)33.33% = 1,733.33%
And using the equation to find the EAR:
EAR = [1 + (APR / m)]m 1
EAR = [1 + .3333]52 1 = 313,916,515.69%
3. The amount of principal paid on the loan is the PV of the monthly payments you make. So,
the present value of the $1,200 monthly payments is:
PVA = $1,200[(1 {1 / [1 + (.068/12)]}360) / (.068/12)] = $184,070.20
The monthly payments of $1,200 will amount to a principal payment of $184,070.20. The
amount of principal you will still owe is:
$250,000 184,070.20 = $65,929.80
This remaining principal amount will increase at the interest rate on the loan until the
end of the loan period. So the balloon payment in 30 years, which is the FV of the
remaining principal will be:
Balloon payment = $65,929.80[1 + (.068/12)]360 = $504,129.05
Chapter 5 problems
4. a. The IRR is the interest rate that makes the NPV of the project equal to zero. So, the
IRR for each project is:
Deepwater Fishing IRR:
0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3
0 = $750,000 + $310,000 / (1 + IRR) + $430,000 / (1 + IRR)2 + $330,000 / (1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the
equation, we find that:
IRR = 19.83%
Submarine Ride IRR:
0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3
0 = $2,100,000 + $1,200,000 / (1 + IRR) + $760,000 / (1 + IRR)2 + $850,000 / (1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the
equation, we find that:
IRR = 17.36%
Based on the IRR rule, the deepwater fishing project should be chosen because it has the higher
IRR.
b.
To calculate the incremental IRR, we subtract the smaller projects cash flows from the larger
projects cash flows. In this case, we subtract the deepwater fishing cash flows from the submarine
ride cash flows. The incremental IRR is the IRR of these incremental cash flows. So, the
incremental cash flows of the submarine ride are:
Submarine Ride
Deepwater Fishing
Submarine Fishing
Year 0
$2,100,000
750,000
$1,350,000
Year 1
$1,200,000
310,000
$890,000
Year 2
$760,000
430,000
$330,000
Year 3
$850,000
330,000
$520,000
Setting the present value of these incremental cash flows equal to zero, we find the incremental
IRR is:
0 = C0 + C1 / (1 + IRR) + C2 / (1 + IRR)2 + C3 / (1 + IRR)3
0 = $1,350,000 + $890,000 / (1 + IRR) + $330,000 / (1 + IRR)2 + $520,000 / (1 + IRR)3
Using a spreadsheet, financial calculator, or trial and error to find the root of the
equation, we find that:
Incremental IRR = 15.78%
For investing-type projects, accept the larger project when the incremental IRR is greater than the
discount rate. Since the incremental IRR, 15.78%, is greater than the required rate of return of 14
percent, choose the submarine ride project. Note that this is not the choice when evaluating only
the IRR of each project. The IRR decision rule is flawed because there is a scale problem. That is,
the submarine ride has a greater initial investment than does the deepwater fishing project. This
problem is corrected by calculating the IRR of the incremental cash flows, or by evaluating the
NPV of each project.
c.
The NPV is the sum of the present value of the cash flows from the project, so the NPV of each
project will be:
Deepwater fishing:
NPV = $750,000 + $310,000 / 1.14 + $430,000 / 1.142 + $330,000 / 1.143
NPV = $75,541.46
Submarine ride:
NPV = $2,100,000 + $1,200,000 / 1.14 + $760,000 / 1.142 + $850,000 / 1.143
NPV = $111,152.69
Since the NPV of the submarine ride project is greater than the NPV of the deepwater fishing
project, choose the submarine ride project. The incremental IRR rule is always consistent with the
NPV rule.
Chapter 6
5. We need to find the bid price for a project, but the project has extra cash flows. Since we
dont already produce the keyboard, the sales of the keyboard outside the contract are
relevant cash flows. Since we know the extra sales number and price, we can calculate the
cash flows generated by these sales. The cash flow generated from the sale of the
keyboard outside the contract is:
Year 1
Year 2
Year 3
Year 4
Sales
$1,100,000 $3,300,000 $3,850,000 $1,925,000
Variable costs
660,000 1,980,000 2,310,000 1,155,000
EBT
$440,000 $1,320,000 $1,540,000 $770,000
Tax
176,000
528,000
616,000
308,000
Net income
(and OCF)
$264,000 $792,000 $924,000 $462,000
So, the addition to NPV of these market sales is:
NPV of market sales = $264,000/1.13 + $792,000/1.132 + $924,000/1.133 +
$462,000/1.134
NPV of market sales = $1,777,612.09
You may have noticed that we did not include the initial cash outlay, depreciation, or fixed
costs in the calculation of cash flows from the market sales. The reason is that it is
irrelevant whether we include these here. Remember that we are not only trying to
determine the bid price, but we are also determining whether the project is feasible. In
other words, we are trying to calculate the NPV of the project, not just the NPV of the bid
price. We will include these cash flows in the bid price calculation. The reason we stated
earlier that whether we included these costs in this initial calculation was irrelevant is
that you will come up with the same bid price if you include these costs in this calculation,
or if you include them in the bid price calculation.
Next, we need to calculate the aftertax salvage value, which is:
Aftertax salvage value = $200,000(1 .40) = $120,000
Instead of solving for a zero NPV as is usual in setting a bid price, the company president
requires an NPV of $100,000, so we will solve for a NPV of that amount. The NPV equation
for this project is (remember to include the NWC cash flow at the beginning of the project,
and the NWC recovery at the end):
NPV = $100,000 = $3,200,000 75,000 + 1,777,612.09 + OCF (PVIFA13%,4) + [($120,000
+
75,000) / 1.134]
Solving for the OCF, we get:
OCF = $1,477,790.75 / PVIFA13%,4 = $496,824.68
Now we can solve for the bid price as follows:
OCF = $496,824.68 = [(P v)Q FC ](1 tC) + tCD
$471,253.44 = [(P $165)(9,000) $600,000](1 0.40) + 0.40($3,200,000/4)
P = $264.41
Chapter 7
a.
To calculate the accounting breakeven, we first need to find the depreciation for each year. The
depreciation = is:
Depreciation $724,000/8
Depreciation = $90,500 per year
And the accounting breakeven is:
QA = ($850,000 + 90,500)/($39 23)
QA = 58,781 units
b.
We will use the tax shield approach to calculate the OCF. The OCF is:
OCFbase = [(P v)Q FC](1 tc) + tcD
OCFbase = [($39 23)(75,000) $850,000](0.65) + 0.35($90,500)
OCFbase = $259,175
Now we can calculate the NPV using our base-case projections. There is no salvage value or
NWC, so the NPV is:
NPVbase = $724,000 + $259,175(PVIFA15%,8)
NPVbase = $439,001.55
To calculate the sensitivity of the NPV to changes in the quantity sold, we will calculate the NPV
at a different quantity. We will use sales of 80,000 units. The NPV at this sales level is:
OCFnew = [($39 23)(80,000) $850,000](0.65) + 0.35($90,500)
OCFnew = $311,175
And the NPV is:
NPVnew = $724,000 + $311,175(PVIFA15%,8)
NPVnew = $672,342.27
So, the change in NPV for every unit change in sales is:
NPV/S = ($439,001.55 672,342.27)/(75,000 80,000)
NPV/S = +$46.668
If sales were to drop by 500 units, then NPV would drop by:
NPV drop = $46.668(500) = $23,334.07
You may wonder why we chose 80,000 units. Because it doesnt matter! Whatever sales number
we use, when we calculate the change in NPV per unit sold, the ratio will be the same.
c.
To find out how sensitive OCF is to a change in variable costs, we will compute the OCF at a
variable cost of $24. Again, the number we choose to use here is irrelevant: We will get the same
ratio of OCF to a one dollar change in variable cost no matter what variable cost we use. So, using
the tax shield approach, the OCF at a variable cost of $24 is:
OCFnew = [($39 24)(75,000) 850,000](0.65) + 0.35($90,500)
OCFnew = $210,425
So, the change in OCF for a $1 change in variable costs is:
OCF/v = ($259,175 210,425)/($23 24)
OCF/v = $48,750
If variable costs decrease by $1 then, OCF would increase by $48,750
Chapter 11
7.
First, we need to find the standard deviation of the market and the portfolio, which are:
M = (.0429)1/2
M = .2071 or 20.71%
Z = (.1783)1/2
Z = .4223 or 42.23%
Now we can use the equation for beta to find the beta of the portfolio, which is:
Z = (Z,M)(Z) / M
Z = (.39)(.4223) / .2071
Z = .80
Now, we can use the CAPM to find the expected return of the portfolio, which is:
E(RZ) = Rf + Z[E(RM) Rf]
E(RZ) = .048 + .80(.114 .048)
E(RZ) = .1005 or 10.05%
Chapter 13
8.
We will begin by finding the market value of each type of financing. We find:
MVD = 5,000($1,000)(1.03) = $5,150,000
MVE = 160,000($57) = $9,120,000
And the total market value of the firm is:
V = $5,150,000 + 9,120,000 = $14,270,000
Now, we can find the cost of equity using the CAPM. The cost of equity is:
RE = .06 + 1.10(.07) = .1370 or 13.70%
The cost of debt is the YTM of the bonds, so:
P0 = $1,030 = $40(PVIFAR%,40) + $1,000(PVIFR%,40)
R = 3.851%
YTM = 3.851% 2 = 7.70%
And the aftertax cost of debt is:
RD = (1 .35)(.0770) = .0501 or 5.01%
Now we have all of the components to calculate the WACC. The WACC is:
WACC = .0501(5.15/14.27) + .1370(9.12/14.27) = .1056 or 10.56%
Notice that we didnt include the (1 tC) term in the WACC equation. We simply used the aftertax cost
of debt in the equation, so the term is not needed here.
Chapter 16
9. a. In a world with corporate taxes, a firms weighted average cost of capital is equal to:
RWACC = [B / (B+S)](1 tC)RB + [S / (B+S)]RS
We do not have the companys debttovalue ratio or the equitytovalue ratio, but
we can calculate either from the debttoequity ratio. With the given debtequity
ratio, we know the company has 2.5 dollars of debt for every dollar of equity. Since
we only need the ratio of debttovalue and equitytovalue, we can say:
B / (B+S) = 2.5 / (2.5 + 1) = .7143
S / (B+S) = 1 / (2.5 + 1) = .2857
We can now use the weighted average cost of capital equation to find the cost of
equity, which is:
.15 = (.7143)(1 0.35)(.10) + (.2857)(RS)
RS = .3625 or 36.25%
b. We can use ModiglianiMiller Proposition II with corporate taxes to find the
unlevered cost of equity. Doing so, we find:
c.
RS = R0 + (B/S)(R0 RB)(1 tC)
.3625 = R0 + (2.5)(R0 .10)(1 .35)
R0 = .2000 or 20.00%
We first need to find the debttovalue ratio and the equitytovalue ratio. We can
then use the cost of levered equity equation with taxes, and finally the weighted
average cost of capital equation. So:
If debtequity = .75
B / (B+S) = .75 / (.75 + 1) = .4286
S / (B+S) = 1 / (.75 + 1) = .5714
The cost of levered equity will be:
RS = R0 + (B/S)(R0 RB)(1 tC)
RS = .20 + (.75)(.20 .10)(1 .35)
RS = .2488 or 24.88%
And the weighted average cost of capital will be:
RWACC = [B / (B+S)](1 tC)RB + [S / (B+S)]RS
RWACC = (.4286)(1 .35)(.10) + (.5714)(.2488)
RWACC = .17
If debtequity =1.50
B / (B+S) = 1.50 / (1.50 + 1) = .6000
E / (B+S) = 1 / (1.50 + 1) = .4000
The cost of levered equity will be:
RS = R0 + (B/S)(R0 RB)(1 tC)
RS = .20 + (1.50)(.20 .10)(1 .35)
RS = .2975 or 29.75%
And the weighted average cost of capital will be:
RWACC = [B / (B+S)](1 tC)RB + [S / (B+S)]RS
RWACC = (.6000)(1 .35)(.10) + (.4000)(.2975)
RWACC = .1580 or 15.80%
Chapter 17
10. a. The total value of a firms equity is the discounted expected cash flow to the firms
stockholders. If the expansion continues, each firm will generate earnings before
interest and taxes of $2.4 million. If there is a recession, each firm will generate
earnings before interest and taxes of only $900,000. Since Steinberg owes its
bondholders $800,000 at the end of the year, its stockholders will receive $1.6
million (= $2,400,000 800,000) if the expansion continues. If there is a recession,
its stockholders will only receive $100,000 (= $900,000 800,000). So, assuming a
discount rate of 15 percent, the market value of Steinbergs equity is:
SSteinberg = [.80($1,600,000) + .20($100,000)] / 1.15 = $1,130,435
Steinbergs bondholders will receive $800,000 whether there is a recession or a
continuation of the expansion. So, the market value of Steinbergs debt is:
BSteinberg = [.80($800,000) + .20($800,000)] / 1.15 = $695,652
Since Dietrich owes its bondholders $1.1 million at the end of the year, its
stockholders will receive $1.3 million (= $2.4 million 1.1 million) if the expansion
continues. If there is a recession, its stockholders will receive nothing since the firms
bondholders have a more senior claim on all $800,000 of the firms earnings. So, the
market value of Dietrichs equity is:
SDietrich = [.80($1,300,000) + .20($0)] / 1.15 = $904,348
Dietrichs bondholders will receive $1.1 million if the expansion continues and
$900,000 if there is a recession. So, the market value of Dietrichs debt is:
BDietrich = [.80($1,100,000) + .20($900,000)] / 1.15 = $921,739
b. The value of company is the sum of the value of the firms debt and equity. So, the
value of Steinberg is:
VSteinberg = B + S
VSteinberg = $1,130,435 + $695,652
VSteinberg = $1,826,087
And value of Dietrich is:
VDietrich = B + S
VDietrich = $904,348 + 921,739
VDietrich = $1,826,087
You should disagree with the CEOs statement. The risk of bankruptcy per se does not
affect a firms value. It is the actual costs of bankruptcy that decrease the value of a
firm. Note that this problem assumes that there are no bankruptcy costs.