Chapter 18: Valuation and Capital Budgeting for the Levered Firm
18.1
Instructor’s Note: For PV(CCA tax shield) formula, refer to pages 238-239. In exams, I would
provide you the formula if necessary.
a.
The maximum price that Budget should be willing to pay for the fleet of cars with all–equity funding
is the price that makes the NPV of the transaction equal to zero.
NPV = –Purchase Price + PV[(1– TC )(Earnings Before Taxes and Depreciation)] +
PV(CCA Tax Shield)
Let P equal the purchase price of the fleet.
NPV = –P + (1–0.38)($430,000)A50.09875 + PVCCATS
 P x 0.38 x 0.25  1  0.5 x 0.09875 
PVCCATS  
 0.26016 P
 0.25  0.09875   1  0.09875 
Set the NPV equal to zero.
0 = –P + (1–0.38)($430,000)A50.09875 + 0.26016P
P = $1,370,376.43
Therefore, the most that Budget should be willing to pay for the fleet of cars with all–equity
funding is $1,370,376.43.
b.
The adjusted present value (APV) of a project equals the net present value of the project if it
were funded completely by equity plus the net present value of any financing side effects. In
Budget’s case, the NPV of financing side effects equals the after–tax present value of the
cash flows resulting from the firm’s debt.
APV = NPV(All–Equity) + NPV(Financing Side Effects)
NPV(All–Equity)
NPV = –Purchase Price + PV[(1– TC )(Earnings Before Taxes and Depreciation)] +
PV(CCATS)
Budget paid $1,100,000 for the fleet of cars.
1,100,000 x 0.38 x 0.25  1  0.5 x 0.09875 
PVCCATS  
  1  0.09875   $286,176.45
0.25  0.09875

NPV = –$1,100,000 + (1– 0.38)($430,000)A50.09875 + $286,176.45
= $200,036
NPV(Financing Side Effects)
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Assume the debt is repaid at the end of the project. The net present value of financing side
effects equals the after–tax present value of cash flows resulting from the firm’s debt.
NPV(Financing Side Effects)
= Proceeds – After–Tax PV(Interest Payments)
– PV(Principal Payments)
Given a known level of debt, debt cash flows should be discounted at the pre–tax cost of debt
(rB), 7%.
NPV(Financing Side Effects)
= $850,000 – (1 – 0.38)(0.07)($850,000)A50.07
– [$850,000/(1.07)5] = $92,705
APV
APV = NPV(All–Equity) + NPV(Financing Side Effects)
= $200,036+ $92,705
= $292,741
Therefore, if Budget uses $850,000 of five–year, 7% debt to fund the $1,100,000 purchase,
the Adjusted Present Value (APV) of the project is $292,741.
c.
18.4
To determine the maximum price, set the APV=0 = NPV (All equity) + NPV(Loan)
0 = –P + (1–0.38)($430,000)A50.09875 + 0.26016P + $850,000
– (1 – 0.38)(0.07)($850,000)A50.07 – [$850,000/(1.07)5]
P = $1,495,680.55
The adjusted present value of a project equals the net present value of the project under all–equity
financing plus the net present value of any financing side effects. First, we need to calculate the
unlevered cost of equity. According to Modigliani–Miller Proposition II with corporate taxes:
rS = r0 + (B/S)( r0 – rB)(1 – Tc)
0.1575= r0 + (0.45)( r0 – 0.088)(1 – 0.40)
r0 = 0.1427 or 14.27%
Now we can find the NPV of an all–equity project, which is:
NPV = PV(Unlevered Cash Flows)
NPV = –$27,000,000 + $9,000,000/1.1427 + $15,000,000/1.14272 + $12,000,000/1.14273
NPV = $405,996.34
Next, we need to find the net present value of financing side effects. This is equal the aftertax
present value of cash flows resulting from the firm’s debt. So:
NPV = Proceeds – Aftertax PV(Interest Payments) – PV(Principal Payments)
Each year, and equal principal payment will be made, which will reduce the interest accrued during
the year. The outstanding balance of the debt for years 2 & 3 are 8,490,000 (=14m-5.51m), and
2,980,000 (=14m-5.1m-5.1m). Given a known level of debt, debt cash flows should be discounted at
the pre–tax cost of debt, so the NPV of the financing effects are:
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NPV = $14,000,000 – (1 – 0.40)(0.088)($14,000,000) / (1.088) – $5,510,000/(1.088)
– (1 –0.40)(0.088)($8,490,000)/(1.088)2 – $5,510,000/(1.088)2
– (1 – 0.40)(0.088)($2,980,000)/(1.088)3 – $2,980,000/(1.088)3
NPV = $786,847.63
So, the APV of project is:
APV = NPV(All–equity) + NPV(Financing side effects)
APV = $405,996.34+ $786,847.63
APV = $1,192,843.97
18.7
a.
In order to value a firm’s equity using the flow–to–equity approach, discount the cash flows
available to equity holders at the cost of the firm’s levered equity. The cash flows to equity
holders will be the firm’s net income. Remembering that the company has three stores, we
find:
One Restaurant
Torino Pizza Club
Sales
$1,000,000
$3,000,000
COGS
450,000
1,350,000
G & A costs
325,000
975,000
Interest
29,500
88,500
EBT
195,500
586,500
Taxes (36%)
70,380
211,140
NI
125,120
$375,360
Since this cash flow will remain the same forever, the present value of cash flows available to
the firm’s equity holders is a perpetuity. We can discount at the levered cost of equity, so, the
value of the company’s equity is:
PV(Flow–to–equity) = $375,360 / 0.19
PV(Flow–to–equity) = $1,975,578.95
b.
The value of a firm is equal to the sum of the market values of its debt and equity, or:
VL= B + S
We calculated the value of the company’s equity in part a, so now we need to calculate the
value of debt. The company has a debt–to–equity ratio of 0.40, which can be written
algebraically as:
B / S = 0.40
We can substitute the value of equity and solve for the value of debt, doing so, we find:
B / $1,975,578.95= 0.40
B = $790,231.58
So, the value of the company is:
V = $1,975,578.95 + $790,231.58
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V = $2,765,810.53
18.8
c.
FTE uses levered cash flow and other methods use unlevered cash flow.
a.
In order to determine the cost of the firm’s debt, we need to find the yield to maturity on its
current bonds. With annual coupon payments, the yield to maturity in the company’s
bonds are:
$975 = $90 A20r + $1,000/(1+r)20
r =YTM= 0.09 or 9%
$984= $105A20r + $1,000/(1+r)20
r=YTM=0.11 or 11%
The cost of the firm’ debt is the weighted average yield to maturity on both bonds:
9% x 975/(975+984) + 11% x 984/(975+984) = 10%
b.
We can use the Capital Asset Pricing Model to find the return on unlevered equity. According
to the Capital Asset Pricing Model:
r0 = rF + βUnlevered(rM – rF)
r0 = 0.07 + 1.1(0.13 – .07)
r0 = 0.1360 or 13.60%
Now we can find the cost of levered equity. According to Modigliani–Miller Proposition II
with corporate taxes
rS = r0 + (B/S)( r0 – rB)(1 – Tc)
rS = 0.1360 + (0.36)(0.1360 – 0.10)(1 – 0.36)
rS = 0.1443 or 14.43%
c.
In a world with corporate taxes, a firm’s weighted average cost of capital is equal to:
rWACC = [B / (B + S)](1 – Tc) rB + [S / (B + S)] rS
The problem does not provide either the debt–value ratio or equity–value ratio. However, the
firm’s debt–equity ratio of is:
B/S = 0.36
Solving for B:
B = 0.36S
Substituting this in the debt–value ratio, we get:
B/V = 0.36S / (0.36S + S)
B/V = 0.36 / 1.36
B/V = 0.265
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And the equity–value ratio is one minus the debt–value ratio, or:
S/V = 1 – 0.265
S/V = 0.735
So, the WACC for the company is:
rWACC = 0.265(1 – 0.36)(0.10) + 0.735(0.1443)
rWACC = 0.1230 or 12.30%
18.9
Whether the company issues stock or issues equity to finance the project is irrelevant. The
company’s optimal capital structure determines the WACC. In a world with corporate taxes,
a firm’s weighted average cost of capital equals:
rWACC = [B / (B + S)](1 – Tc) rB + [S / (B + S)] rS
rWACC = 0.80(1 – 0.34)(0.072) + 0.20(0.1090)
rWACC = 0.0598 or 5.98%
Now we can use the weighted average cost of capital to discount NEC’s unlevered cash
flows. Doing so, we find the NPV of the project is:
NPV = –$50,000,000 + $3,500,000 / 0.0598
NPV = $8,528,428.09
Yes, accept the project.
18.12. a.
Since the company is currently an all–equity firm, its value equals the present value of its
unlevered after–tax earnings, discounted at its unlevered cost of capital. The cash flows to
shareholders for the unlevered firm are:
EBIT
Tax
Net income
$75,000
30,000
$45,000
So, the value of the company is:
VU = $45,000 / 0.18
VU = $250,000
b.
The adjusted present value of a firm equals its value under all–equity financing plus the net
present value of any financing side effects. In this case, the NPV of financing side effects
equals the after–tax present value of cash flows resulting from debt. Given a known level of
debt, debt cash flows should be discounted at the pre–tax cost of debt, so:
NPV = Proceeds – Aftertax PV(Interest payments)
NPV = $160,000 – (1 – 0.40)(0.10)($160,000) / 0.10
NPV = $64,000
So, using the APV method, the value of the company is:
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APV = VU + NPV(Financing side effects)
APV = $250,000 + 64,000
APV = $314,000
The value of the debt is given, so the value of equity is the value of the company minus the
value of the debt, or:
S=V–B
S = $314,000 – $160,000
S = $154,000
c.
According to Modigliani–Miller Proposition II with corporate taxes, the required return of
levered equity is:
rS= r0 + (B/S)( r0 – rB)(1 – Tc)
rS = 0.18 + ($160,000 / $154,000)(0.18 – 0.10)(1 – 0.40)
rS = 0.2299 or 22.99%
d.
In order to value a firm’s equity using the flow–to–equity approach, we can discount the cash
flows available to equity holders at the cost of the firm’s levered equity. First, we need to
calculate the levered cash flows available to shareholders, which are:
EBIT
Interest
EBT
Tax
Net income
$75,000
16,000
$59,000
23,600
$35,400
So, the value of equity with the flow–to–equity method is:
S = Cash flows available to equity holders / rS
S = $35,400 / 0.2299
S = $153,980
(Note that the answer is slightly different from answer in (a). This is due to rounding error.)
18.14
a.
Assume no fixed costs. If the company were financed entirely by equity, the value of the firm
would be equal to the present value of its unlevered after–tax earnings, discounted at its
unlevered cost of capital. First, we need to find the company’s unlevered cash flows, which
are:
Sales
$23,500,000
Variable costs 14,100,000
EBT
$9,400,000
Tax
3,760,000
Net income
$5,640,000
So, the value of the unlevered company is:
VU= $5,640,000 / 0.17
VU= $33,176,470.59
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b.
According to Modigliani–Miller Proposition II with corporate taxes, the value of levered
equity is:
rS = r0 + (B/S)( r0 – rB)(1 – Tc)
rS = 0.17 + (0.45)(0.17 – 0.09)(1 – 0.40)
rS = 0.1916 or 19.16%
c.
In a world with corporate taxes, a firm’s weighted average cost of capital equals:
rWACC = [B / (B + S)](1 – Tc)rB + [S / (B + S)] rS
So we need the debt–value and equity–value ratios for the company. The debt–equity ratio for
the company is:
B/S = 0.45
B = 0.45S
Substituting this in the debt–value ratio, we get:
B/V = 0.45S / (0.45S + S)
B/V = 0.45 / 1.45
B/V = 0.31
And the equity–value ratio is one minus the debt–value ratio, or:
S/V = 1 – 0.31
S/V = 0.69
So, using the capital structure weights, the company’s WACC is:
rWACC = [B / (B + S)](1 – Tc) rB + [S / (B + S)]rS
rWACC = 0.31(1 – 0.40)(0.09) + 0.69(0.1916)
rWACC = 0.1489 or 14.89%
We can use the weighted average cost of capital to discount the firm’s unlevered aftertax
earnings to value the company. Doing so, we find:
VL= $5,640,000 / 0.1489
VL= $37,877,770.32
Now we can use the debt–value ratio and equity–value ratio to find the value of debt and
equity, which are:
B = VL (Debt/ Value)
B = $37,877,770.32 (0.31)
B = $11,742,108.88
S = VL (Equity/value)
S = $37,877,770.32 (0.69)
S = $26,135,661.52
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d.
In order to value a firm’s equity using the flow–to–equity approach, we can discount the cash
flows available to equity holders at the cost of the firm’s levered equity. First, we need to
calculate the levered cash flows available to shareholders, which are:
Sales
$23,500,000
Variable costs 14,100,000
EBIT
$9,400,000
Interest
1,056,790
EBT
$8,343,210
Tax
3,337,284
Net income
$5,005,926
So, the value of equity with the flow–to–equity method is:
S = Cash flows available to equity holders / rS
S = $5,005,926 / 0.1916
S = $26,126,962.42
(Again, some minor difference from (c) due to rounding error.)
18.16
e.
The WACC is based on a target debt level while the APV is based on the amount of debt.
a.
If flotation costs are not taken into account, the net present value of a loan equals:
NPVLoan = Gross Proceeds – Aftertax present value of interest and principal payments
NPVLoan = $5,312,500 – .084($5,312,500)(1 – 0.31) A120.084 – $5,312,500/1.08412
NPVLoan = $1,021,256.98
b.
The floatation costs of the loan will be:
Floatation costs = $5,312,500 (0.0075)
Floatation costs = $39,843.75
So, the annual floatation expense will be:
Annual floatation expense = $39,843.75/ 12
Annual floatation expense = $3320.31
If flotation costs are taken into account, the net present value of a loan equals:
NPVLoan = Proceeds net of flotation costs – Aftertax present value of interest and principal
payments + Present value of the flotation cost tax shield
NPVLoan = ($5,312,500 – $39,843.75) – 0.084($5,312,500)(1 – 0.31) A120.084
– $5,312,500/1.08412+ ($3,320.31x.0.31) A120.084
NPVLoan = $989,011.87
18.17
a.
You can estimate the unlevered beta from a levered beta. The unlevered beta is the beta of the
assets of the firm; as such, it is a measure of the business risk. Note that the unlevered beta
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will always be lower than the levered beta (assuming the betas are positive). The difference is
due to the leverage of the company. Thus, the second risk factor measured by a levered beta
is the financial risk of the company.
b.
The equity beta of a firm financed entirely by equity is equal to its unlevered beta. Since each
firm has an unlevered beta of 1.25, we can find the equity beta for each. Doing so, we find:
North Pole
βEquity = [1 + (1 – Tc)(B/S)] β Unlevered
βEquity = [1 + (1 – 0.35)($1,400,000/$2,600,000](1.25)
βEquity = 1.69
South Pole
βEquity = [1 + (1 – Tc)(B/S)] β Unlevered
βEquity = [1 + (1 – 0.35)($2,600,000/$1,400,000](1.25)
βEquity = 2.76
c.
We can use the Capital Asset Pricing Model to find the required return on each firm’s equity.
Doing so, we find:
North Pole:
rS= rF + βEquity (rM – rF)
rS = .0530 + 1.69(0.1240 – .0530)
rS = 0.1728 or 17.28%
South Pole:
rS = rF + βEquity (rM – rF)
rS = .0530 + 2.76(0.1240 – .0530)
rS = 0.2489 or 24.89%
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