Finance Chapter 18 Homework The Npv Allequity Project Would Remain Unchanged

CHAPTER 18

VALUATION AND CAPITAL

BUDGETING FOR THE LEVERED FIRM

Answers to Concepts Review and Critical Thinking Questions

2. The WACC is based on a target debt level while the APV is based on the amount of debt.

4. The WACC method does not explicitly include the interest cash flows, but it does implicitly include

5. 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 will always be

Solutions to Questions and Problems

NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple

steps. Due to space and readability constraints, when these intermediate steps are included in this solutions

manual, rounding may appear to have occurred. However, the final answer for each problem is found

without rounding during any step in the problem.

1. a. The maximum price that the company 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. Discounting the

depreciation tax shield at the risk-free rate, the NPV equation for the project is:

NPV = –Purchase Price + PV[(1 – TC)(EBTD)] + PV(Depreciation Tax Shield)

If we let P equal the purchase price of the fleet, then the NPV is:

Setting the NPV equal to zero and solving for the purchase price, we find:

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 this case,

the NPV of financing side effects equals the after-tax present value of the cash flows resulting

from the firm’s debt, so:

NPV(All-Equity)

NPV = –Purchase Price + PV[(1 – TC)(EBTD)] + PV(Depreciation Tax Shield)

The company paid $675,000 for the fleet of cars. Because this fleet will be fully depreciated over

five years using the straight-line method, annual depreciation expense equals:

NPV(Financing Side Effects)

The net present value of financing side effects equals the after-tax present value of cash flows

resulting from the firm’s debt, so:

NPV = Proceeds – Aftertax 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.

So, the NPV of the financing side effects are:

2. 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 this case, the

NPV of financing side effects equals the after-tax present value of the cash flows resulting from the

firm’s debt, so:

APV = NPV(All-Equity) + NPV(Financing Side Effects)

So, the NPV of each part of the APV equation is:

NPV(All-Equity)

NPV(Financing Side Effects)

The net present value of financing side effects equals the aftertax present value of cash flows resulting

from the firm’s debt. So, the NPV of the financing side effects is:

NPV = Proceeds(Net of flotation) – Aftertax PV(Interest Payments) – PV(Principal Payments)

+ PV(Flotation Cost Tax Shield)

Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt, RB.

Since the flotation costs will be amortized over the life of the loan, the annual flotation costs that will

be expensed each year are:

3. 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:

Sales

$3,825,000

COGS

2,235,000

G & A costs

1,215,000

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:

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 .40, which can be written algebraically as:

4. 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 semiannual coupon payments, the yield to maturity of the company’s bonds

is:

Interest

EBT

Taxes

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 = .035 + .95(.11 – .035)

R0 = .1063, or 10.63%

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 is:

5. a. The equity beta of a firm financed entirely by equity is equal to its unlevered beta. Since each

firm has an unlevered beta of 1.05, we can find the equity beta for each. Doing so, we find:

North Pole

βEquity = [1 + (1 – TC)(B/S)]βUnlevered

βEquity = [1 + (1 – .21)($2,400,000/$4,100,000](1.05)

b. 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)

South Pole:

RS = RF + βEquity(RM – RF)

6. 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 = $4,600,000 – .063($4,600,000)(1 – .21)PVIFA6.3%,10 – $4,600,000/1.06310

NPVLoan = $441,621.98

b. The flotation costs of the loan will be:

7. First we need to find the aftertax value of the revenues minus expenses. The aftertax value is:

Aftertax revenue = $3,800,000(1 – .23)

Aftertax revenue = $2,926,000

Next, we need to find the depreciation tax shield. The depreciation tax shield each year is:

8. 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:

9. a. The company has a capital structure with three parts: long-term debt, short-term debt, and equity.

Since interest payments on both long-term and short-term debt are tax-deductible, multiply the

pretax costs by (1 – TC) to determine the aftertax costs to be used in the weighted average cost of

capital calculation. The WACC using the book value weights is:

RWACC = (XSTD)(RSTD)(1 – TC) + (XLTD)(RLTD)(1 – TC) + (XEquity)(REquity)

b. Using the market value weights, the company’s WACC is:

c. Using the target debt-equity ratio, the target debt-value ratio for the company is:

B/S = .60

B = .6S

Substituting this in the debt-value ratio, we get:

And the equity-value ratio is one minus the debt-value ratio, or:

S/V = 1 – .375

S/V = .625

We can use the ratio of short-term debt to long-term debt in a similar manner to find the short-

term debt to total debt and long-term debt to total debt. Using the short-term debt to long-term

debt ratio, we get:

And the long-term debt to total debt ratio is one minus the short-term debt to total debt ratio, or:

LTD/B = 1 – .167

LTD/B = .833

Now we can find the short-term debt to value ratio and long-term debt to value ratio by

multiplying the respective ratio by the debt-value ratio. So:

And the long-term debt to value ratio is:

d. The differences in the WACCs are due to the different weighting schemes. The company’s

WACC will most closely resemble the WACC calculated using target weights since future

projects will be financed at the target ratio. Therefore, the WACC computed with target weights

should be used for project evaluation.

10. 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. In the joint venture’s case, the NPV

of financing side effects equals the aftertax present value of cash flows resulting from the firms’ debt.

So, the APV is:

APV = NPV(All-Equity) + NPV(Financing Side Effects)

11. If the company had to issue debt under the terms it would normally receive, the interest rate on the

debt would increase to the company’s normal cost of debt. The NPV of an all-equity project would

remain unchanged, but the NPV of the financing side effects would change. The NPV of the financing

side effects would be:

12. 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)

.15 = R0 + (.50)(R0 – .09)(1 – .21)

13. a. To calculate the NPV of the project, we first need to find the company’s 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

The market value of the company’s equity is:

Market value of equity = 4,200,000($35)

 = S,M/

2

M

 = .0415/.202

 = 1.04

Now we can use the Capital Asset Pricing Model to determine the cost of equity. The Capital

NPV = $14,904,676.75

b. The weighted average cost of capital used in part a will not change if the firm chooses to fund

the project entirely with debt. The weighted average cost of capital is based on optimal capital

structure weights. Since the current capital structure is optimal, all-debt funding for the project

implies that the firm will have to use more equity in the future to bring the capital structure back

14. We have four companies with comparable operations, so the industry average beta can be used as the

beta for this project. So, the average unlevered beta is:

Unlevered = (1.15 + 1.08 + 1.30 + 1.25)/4

Unlevered = 1.20

A debt-to-value ratio of .35 means that the equity-to-value ratio is .65. This implies a debt-equity ratio

15. a. The company is currently an all-equity firm, so the value as an all-equity firm equals the present

value of aftertax cash flows, discounted at the cost of the firm’s unlevered cost of equity. So, the

current value of the company is:

VU = [(Pretax earnings)(1 – TC)]/R0

VU = [($23,500,000)(1 – .21)]/.11

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 aftertax present value of cash flows resulting from the firm’s debt. Given a known level of

debt, debt cash flows can be discounted at the pretax cost of debt, so the NPV of the financing

effects is:

NPV = Proceeds – Aftertax PV(Interest Payments)

NPV = $35,000,000 – (1 – .21)(.06)($35,000,000)/.06

c. The company will use the entire proceeds to repurchase equity. Using the share price we

calculated in part b, the number of shares repurchased will be:

Shares repurchased = $35,000,000/$92.70

The value of the company increased, but part of that increase will be funded by the new debt.

The value of equity after recapitalization is the total value of the company minus the value of

debt, or:

New value of equity = $176,122,727 – 35,000,000

New value of equity = $141,122,727

d. In order to value a firm’s equity using the flow–to-equity approach, we must discount the cash

flows available to equity holders at the cost of the firm’s levered equity. 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 = .11 + ($35,000,000/$141,122,727)(.11 – .06)(1 – .21)

RS = .1198, or 11.98%

After the recapitalization, the net income of the company will be:

EBIT

$23,500,000

Interest

2,100,000

16. a. 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

$18,600,000

Variable costs

11,160,000

b. According to Modigliani-Miller Proposition II with corporate taxes, the value of levered equity

is:

RS = R0 + (B/S)(R0 – RB)(1 – TC)

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 = .35

B = .35S

EBT

$7,440,000

Tax

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 = $5,877,600/.1144

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

$18,600,000

Variable costs

11,160,000

EBIT

$7,440,000

Interest

EBT

$6,600,922

Tax

17. 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

$146,000

Tax

30,660

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 = $325,000 – (1 – .21)(.07)($325,000)/.07

NPV = $68,250

So, using the APV method, the value of the company is:

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)

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

$146,000

Interest

22,750

Net income

18. Since the company is not publicly traded, we need to use the industry numbers to calculate the industry

levered return on equity. We can then find the industry unlevered return on equity, and re-lever the

industry return on equity to account for the different use of leverage. So, using the CAPM to calculate

the industry levered return on equity, we find:

RS = RF + β(MRP)

Next, to find the average cost of unlevered equity in the holiday gift industry we can use Modigliani-

Miller Proposition II with corporate taxes, so:

Now, we can use the Modigliani-Miller Proposition II with corporate taxes to re-lever the return on

equity to account for this company’s debt-equity ratio. Doing so, we find:

Since the project is financed at the firm’s target debt-equity ratio, it must be discounted at the

company’s weighted average cost of capital. In a world with corporate taxes, a firm’s weighted average

cost of capital equals:

S/V = 1 – .29

S/V = .71

So, using the capital structure weights, the company’s WACC is:

RWACC = [B/(B + S)](1 – TC)RB + [S/(B + S)]RS

RWACC = .29(1 – .21)(.05) + .71(.1366)

Now we need the project’s cash flows. The cash flows increase for the first five years before leveling

off into perpetuity. So, the cash flows from the project for the next six years are:

Year 1 cash flow

$93,000.00

Year 2 cash flow

$97,650.00

Year 3 cash flow

Year 4 cash flow

Year 5 cash flow

Year 6 cash flow