berk chapter 18: capital budgeting & leverage

Copyright © 2011 Pearson Prentice Hall. All rights reserved.

Chapter 18

Capital Budgeting and Valuation with Leverage

Copyright © 2011 Pearson Prentice Hall. All rights reserved.18-2

Chapter Outline

18.1 Overview18.2

The Weighted Average Cost of Capital Method

18.3 The Adjusted Present Value Method18.4 The Flow-to-Equity Method18.5 Project-Based Costs of Capital18.6 APV with Other Leverage Policies


Chapter Outline (cont'd)

18.7 Other Effects of Financing18.8 Advanced Topics in Capital Budgeting


Learning Objectives

1. Describe three methods of valuation discussed in the chapter, and list the steps in computing each.

2. Compute the unlevered and equity costs of capital, and explain how they are related.

3. Estimate the cost of capital for a project, even if its risk is different from that of the firm as a whole.

4. Estimate the cost of capital for a project, given the project’s debt-to-value ratio, assuming (1) the firm maintains a target leverage ratio, or (2) some tax shields are predetermined.

5. Discuss the importance of considering the incremental impact of the leverage of a project on the firm overall.


Learning Objectives (cont'd)

6. Calculate the levered value of a project if (1) the firm has a constant interest coverage policy, or (2) the firm keeps debt at a constant level.

7. Describe situations in which the WACC method is best to use and situations in which the APV method is advisable.

8. Discuss how issuance costs and mispricing costs should be included in the assessment of the project’s value.

9. Calculate the value of the interest tax shield if a firm adjusts its debt annually to a target level.

10. Describe the effects of financial distress on the use of leverage.

11. Adjust the APV method for personal taxes.


18.1 Overview

• Assumptions in this chapter– The project has average risk.– The firm’s debt-equity ratio is constant.– Corporate taxes are the only imperfection.


18.2 The Weighted Average Cost of Capital Method• For now, it is assumed that the firm

maintains a constant debt-equity ratio and that the WACC remains constant over time.


18.2 The Weighted Average Cost of Capital Method (cont'd)

– Because the WACC incorporates the tax savings from debt, we can compute the levered value of an investment, by discounting its future free cash flow using the WACC.

(1 ) wacc E D cE Dr r r

E D E D

31 20 2 3

1 (1 ) (1 )L

wacc wacc wacc

FCFFCF FCFVr r r


Using the WACC to Value a Project

• Assume Avco is considering introducing a new line of packaging, the RFX Series.– Avco expects the technology used in these

products to become obsolete after four years. However, the marketing group expects annual sales of $60 million per year over the next four years for this product line.

– Manufacturing costs and operating expenses are expected to be $25 million and $9 million, respectively, per year.


Using the WACC to Value a Project (cont'd)

– Developing the product will require upfront R&D and marketing expenses of $6.67 million, together with a $24 million investment in equipment.

• The equipment will be obsolete in four years and will be depreciated via the straight-line method over that period.

– Avco expects no net working capital requirements for the project.

– Avco pays a corporate tax rate of 40%.


Table 18.1 Spreadsheet Expected Free Cash Flow from Avco’s RFX Project


Table 18.2 Spreadsheet Avco’s Current Market Value Balance Sheet ($ million) and Cost of Capital Without the RFX Project


Using the WACC to Value a Project (cont'd)• Avco intends to maintain a similar (net) debt-

equity ratio for the foreseeable future, including any financing related to the RFX project. Thus, Avco’s WACC is

• Note that net debt = D=320-20=$300 million.

300 300 (1 ) (10%) (6%)(1 0.40) 600 600

6.8%

wacc E D cE Dr r r

E D E D


Using the WACC to Value a Project (cont'd)• The value of the project, including the tax

shield from debt, is calculated as the present value of its future free cash flows.

• The NPV of the project is $33.25 million– $61.25 million – $28 million = $33.25 million

0 2 3 4

18 18 18 18 $61.25 million1.068 1.068 1.068 1.068

LV


Summary of the WACC Method

1. Determine the free cash flow of the investment.

2. Compute the weighted average cost of capital.

3. Compute the value of the investment, including the tax benefit of leverage, by discounting the free cash flow of the investment using the WACC.


Summary of the WACC Method (cont'd)• The WACC can be used throughout the firm

as the companywide cost of capital for new investments that are of comparable risk to the rest of the firm and that will not alter the firm’s debt-equity ratio.


Textbook Example 18.1


Textbook Example 18.1 (cont'd)


Alternative Example 18.1

• Problem– Assume:

• Chittenden is considering the acquisition of another firm in its industry.

• The acquisition is expected to increase Chittenden’s free cash flow by $5 million the first year and this contribution is expected to grow at a rate of 4% per year from then on.

• Chittenden has negotiated a purchase price of $110 million.


Alternative Example 18.1 (cont’d)

• Problem (continued)– Assume:

• Chittenden’s weighted average cost of capital is 7.5%.• After the transaction, Chittenden will adjust its capital

structure to maintain its current debt-equity ratio of 2.

– If the acquisition has similar risk to the rest of Chittenden, what is the value of this deal?



• Solution– The acquisition can be viewed as a growing

perpetuity:

– Note:• Because the acquisition’s risk matches the risk for the

rest of Chittenden, and because Chittenden will maintain the same debt-equity ratio going forward, the WACC will remain at 7.5%.

$5 $142,857,1437.5% 4%

L millionV



• Solution (cont’d)– The NPV of the acquisition is:

• $142,857,143 – $110,000,000 = $32,857,143


Implementing a Constant Debt-Equity Ratio• By undertaking the RFX project, Avco adds

new assets to the firm with initial market value $61.25 million. – Therefore, to maintain its debt-to-value ratio,

Avco must add $30.625 million in new debt.• 50% × 61.25 = $30.625


Implementing a Constant Debt-Equity Ratio (cont'd)• Avco can add this debt either by reducing

cash or by borrowing and increasing debt. – Assume Avco decides to spend its $20 million

in cash and borrow an additional $10.625 million.

• Because only $28 million is required to fund the project, Avco will pay the remaining $2.625 million to shareholders through a dividend (or share repurchase).

– $30.625 million − $28 million = $2.625 million


Table 18.3 Avco’s Current Market Value Balance Sheet ($ million) with the RFX Project


Implementing a Constant Debt-Equity Ratio (cont'd)• The market value of Avco’s equity

increases by $30.625 million.– $330.625 − $300 = $30.625

• Adding the dividend of $2.625 million, the shareholders’ total gain is $33.25 million.– $30.625 + 2.625 = $33.25– Which is exactly the NPV calculated for the

RFX project


Implementing a Constant Debt-Equity Ratio (cont'd)• Debt Capacity

– The amount of debt at a particular date that is required to maintain the firm’s target debt-to-value ratio

– The debt capacity at date t is calculated as:

• Where d is the firm’s target debt-to-value ratio and VLt

is the levered continuation value on date t.

Lt tD d V


Implementing a Constant Debt-Equity Ratio (cont'd)• Debt Capacity

– VLt calculated as:

Value of in year 2 and beyond

1 1 1

FCF t

LL t tt

wacc

FCF VVr


Table 18.4 Spreadsheet Continuation Value and Debt Capacity of the RFX Project over Time



• Problem– Expanding on the previous example, assume

Chittenden proceeds with the acquisition: • How much debt must Chittenden use to finance

the acquisition and still maintain its debt-to-value ratio?

• How much of the acquisition cost must be financed with equity?



• Solution– From the previous example, the market value

of the assets acquired in the acquisition is $142,857,143.

– To maintain its debt-to-equity ratio of 2, Chittenden must increase its debt by $95,242,857.

2 66.67%1 2D

DWE D

$142,857,173 66.67% $95,242,857



• Solution– The remaining $14,757,143 of the

$110,000,000 acquisition cost will be financed with new equity.

– In addition to the $14,757,143 in new equity, Chittenden’s existing shares will increase in value by $32,857,143 (the NPV of the acquisition), for a total increase in equity of $47,614,286.


18.3 The Adjusted Present Value Method• Adjusted Present Value (APV)

– A valuation method to determine the levered value of an investment by first calculating its unlevered value and then adding the value of the interest tax shield.

(Interest Tax Shield) L UV APV V PV


The Unlevered Value of the Project

• The first step in the APV method is to calculate the value of the free cash flows using the project’s cost of capital if it were financed without leverage.


The Unlevered Value of the Project (cont'd)• Unlevered Cost of Capital

– The cost of capital of a firm, were it unlevered: for a firm that maintains a target leverage ratio, it can be estimated as the weighted average cost of capital computed without taking into account taxes (pre-tax WACC).

– We value the interest tax shield separately.

Pretax WACC U E DE Dr r r

E D E D


The Unlevered Value of the Project (cont'd)• The firm’s unlevered cost of capital equals

its pretax WACC because it represents investors’ required return for holding the entire firm (equity and debt).

• This argument relies on the assumption that the overall risk of the firm is independent of the choice of leverage.

• In appendix 18A.2, we show that the tax shield will have the same risk as the firm if the firm maintains a target leverage ratio.


The Unlevered Value of the Project (cont'd)• Target Leverage Ratio

– When a firm adjusts its debt proportionally to a project’s value or its cash flows (where the proportion need not remain constant)

– A constant market debt-equity ratio is a special case.


The Unlevered Value of the Project (cont'd)• For Avco, its unlevered cost of capital is

calculated as:

• The project’s value without leverage is calculated as:

0.50 10.0% 0.50 6.0% 8.0%Ur

2 3 4

18 18 18 18 $59.62 million1.08 1.08 1.08 1.08

UV


Valuing the Interest Tax Shield

• The value of $59.62 million is the value of the unlevered project and does not include the value of the tax shield provided by the interest payments on debt.

• The interest tax shield is equal to the interest paid multiplied by the corporate tax rate.

1Interest paid in year D tt r D


Table 18.5 Spreadsheet Expected Debt Capacity, Interest Payments, and Tax Shield for Avco’s RFX Project


Valuing the Interest Tax Shield (cont'd)• The next step is to find the present value

of the interest tax shield.– When the firm maintains a target leverage

ratio, its future interest tax shields have similar risk to the project’s cash flows, so they should be discounted at the project’s unlevered cost of capital.

2 3 4

0.73 0.57 0.39 0.20(interest tax shield) $1.63 million1.08 1.08 1.08 1.08

PV


Valuing the Interest Tax Shield (cont'd)• The total value of the project with leverage

is the sum of the value of the interest tax shield and the value of the unlevered project.

– The NPV of the project is $33.25 million• $61.25 million – $28 million = $33.25 million

– This is exactly the same value found using the WACC approach.

(interest tax shield) 59.62 1.63 $61.25 millionL UV V PV


Summary of the APV Method

1. Determine the investment’s value without leverage.

2. Determine the present value of the interest tax shield.a. Determine the expected interest tax shield.b. Discount the interest tax shield.

3. Add the unlevered value to the present value of the interest tax shield to determine the value of the investment with leverage.


Summary of the APV Method (cont'd)• The APV method has some advantages.

– It can be easier to apply than the WACC method when the firm does not maintain a constant debt-equity ratio.

– The APV approach also explicitly values market imperfections and therefore allows managers to measure their contribution to value.



• Problem:– Consider again Chittenden’s acquisition from

Alternative Examples 18.1 and 18.2. The acquisition will contribute $5.0 million in free cash flows the first year, which will grow by 4% per year thereafter. The acquisition cost of $110 million will be financed with $95,242,857 in new debt initially. Compute the value of the acquisition using the APV method, assuming Chittenden will maintain a constant debt-equity ratio for the acquisition.


Alternative Example 18.3 (cont'd)

• Solution– First, we compute the value without leverage.

Using Equation 18.6, we first calculate Chittenden’s unlevered cost of capital. Assuming the company’s cost of equity is 12.3%, its cost of debt is 8.5%, and its tax rate is 40%,

1 2(12.3%) (8.5%) 9.77%3 3Ur



• Solution (cont’d)– Given the free cash flow of $5 million and the

growth rate of 4%, we get:VU = 5.0/(9.77% − 4%) = $86.65 million

– Chittenden will add new debt of approximately $95.24 million initially to fund the acquisition. At an 8.5% interest rate, the interest expense the first year is .085 × 95.24= $8.1 million, which provides an interest tax shield of 40% × $8.1 = $3.24 million.



• Solution (cont’d)– Because the value of the acquisition is

expected to grow by 4% per year, the amount of debt the acquisition supports—and, therefore, the interest tax shield—is expected to grow at the same rate. The present value of the interest tax shield is

– PV(interest tax shield) = $3.24/(9.77% − 4%) = $56.15 million.



• Solution (cont’d)– The value of the acquisition with leverage is

given by the APV:VL = VU + PV(interest tax shield) = 86.65 + 56.15 =

$142.8 million– This value is identical (with rounding) to the value

computed in Alternative Example 18.1 and implies an NPV of 142.8-110 = $32.8 million for the acquisition. Without the benefit of the interest tax shield, the NPV would be 86.65 − 80 = $6.65 million.


18.3 The Adjusted Present Value Method (cont’d)• We can easily extend the APV approach to

include other market imperfections such as financial distress, agency, and issuance costs.

• Section 18.7 will elaborate on these complexities.


18.4 The Flow-to-Equity Method

• Flow-to-Equity– A valuation method that calculates the free

cash flow available to equity holders taking into account all payments to and from debt holders

– The cash flows to equity holders are then discounted using the equity cost of capital.


Calculating the Free Cash Flow to Equity• Free Cash Flow to Equity (FCFE)

– The free cash flow that remains after adjusting for interest payments, debt issuance and debt repayments

• The first step in the FTE method is to determine the project’s free cash flow to equity.


Table 18.6 Spreadsheet Expected Free Cash Flows to Equity from Avco’s RFX Project


Calculating the Free Cash Flow to Equity (cont'd)• Note two changes in the calculation of the

free cash flows.– Interest expenses are deducted before taxes. – The proceeds from the firm’s net borrowing

activity are added in.• These proceeds are positive when the firm issues

debt and are negative when the firm reduces its debt by repaying principal.

1Net Borrowing at Date t tt D D


Calculating the Free Cash Flow to Equity (cont'd)• The FCFE can also be calculated, using the

free cash flow, as (1 ) (Interest Payments) (Net Borrowing)cFCFE FCF

After-tax interest expense


Table 18.7 Spreadsheet Computing FCFE from FCF for Avco’s RFX Project


Valuing Equity Cash Flows

• Because the FCFE represent payments to equity holders, they should be discounted at the project’s equity cost of capital.– Given that the risk and leverage of the RFX

project are the same as for Avco overall, we can use Avco’s equity cost of capital of 10.0% to discount the project’s FCFE.

2 3 4

9.98 9.76 9.52 9.27( ) 2.62 $33.25 million1.10 1.10 1.10 1.10

NPV FCFE


Valuing Equity Cash Flows (cont'd)

• The value of the project’s FCFE represents the gain to shareholders from the project and it is identical to the NPV computed using the WACC and APV methods.


Summary of the Flow-to-Equity Method1. Determine the free cash flow to equity of

the investment.2. Determine the equity cost of capital.3. Compute the equity value by discounting

the free cash flow to equity using the equity cost of capital.


Summary of the Flow-to-Equity Method (cont'd)• The FTE method offers some advantages.

– It may be simpler to use when calculating the value of equity for the entire firm, if the firm’s capital structure is complex and the market values of other securities in the firm’s capital structure are not known.

– It may be viewed as a more transparent method for discussing a project’s benefit to shareholders by emphasizing a project’s implication for equity.

• The FTE method has a disadvantage.– One must compute the project’s debt capacity to

determine the interest and net borrowing before capital budgeting decisions can be made.



• Problem:– Consider again Chittenden’s acquisition from

Alternative Examples 18.1 and 18.2. The acquisition will contribute $5.0 million in free cash flows the first year, which will grow by 4% per year thereafter. The acquisition cost of $110 million will be financed with $95,242,857 in new debt initially. What is the value of this acquisition using the FTE method?



• Solution– Because the acquisition is being financed with

$95.24 million in new debt, the remaining $14.76 million of the acquisition cost must come from equity:

FCFE0 = −110 + 95.24 = −$14.76 million– In one year, the interest on the debt will be

8.5% × 95.24 = $8.1 million.



• Solution (cont’d)– Because Chittenden maintains a constant debt-

equity ratio, the debt associated with the acquisition is also expected to grow at a 4% rate: $95.24 × 1.04 = $99.05 million. Therefore, Chittenden will borrow an additional $99.05 - $95.24= $3.81 million in one year.

FCFE1 = +5.0 − (1 − 0.40) × 8.1 + 3.81 = $3.95 million



• Solution (cont’d)– After year 1, FCFE will also grow at a 4% rate.

Using the cost of equity rE = 12.3%, we compute the NPV:

NPV(FCFE ) = −14.76 + 3.95/(12.3% − 4%) = $32.8 million

– This NPV matches the result we obtained with the WACC and APV methods.


18.5 Project-Based Costs of Capital

• In the real world, a specific project may have different market risk than the average project for the firm.

• In addition, different projects will may vary in the amount of leverage they will support.


Estimating the Unlevered Cost of Capital• Suppose Avco launches a new plastics

manufacturing division that faces different market risks than its main packaging business.– The unlevered cost of capital for the plastics

division can be estimated by looking at other single-division plastics firms that have similar business risks.


Estimating the Unlevered Cost of Capital (cont'd)• Assume two firms are comparable to the

plastics division and have the following characteristics:


Estimating the Unlevered Cost of Capital (cont'd)• Assuming that both firms maintain a target

leverage ratio, the unlevered cost of capital for each competitor can be estimated by calculating their pretax WACC.

Competitor 1: 0.60 12.0% 0.40 6.0% 9.6%Competitor 2: 0.75 10.7% 0.25 5.5% 9.4%

U

U

rr


Estimating the Unlevered Cost of Capital (cont'd)• Based on these comparable firms, we

estimate an unlevered cost of capital for the plastics division is approximately 9.5%.

• With this rate in hand we can use APV approach.

• To use WACC or FTE method we need to estimate the project’s equity cost of capital, which depends on the incremental debt the company will take on as a result of the project.


Project Leverage and the Equity Cost of Capital• A project’s equity cost of capital may differ

from the firm’s equity cost of capital if the project uses a target leverage ratio that is different than the firm’s. The project’s equity cost of capital can be calculated as:

( )E U U DDr r r rE


Project Leverage and the Equity Cost of Capital (cont'd)• Now assume that Avco plans to maintain

an equal mix of debt and equity financing as it expands into plastics manufacturing, and it expects its borrowing cost to be 6%.– Given the unlevered cost of capital estimate of

9.5%, the plastics division’s equity cost of capital is estimated to be:

0.50 9.5% (9.5% 6%) 13.0%0.50Er


Project Leverage and the Equity Cost of Capital (cont'd)• The division’s WACC can now be estimated

to be:

• An alternative method for calculating the division’s WACC is:

0.50 13.0% 0.50 6.0% (1 0.40) 8.3%WACCr

wacc wacc c Dr r d r 9.5% 0.50 0.40 6% 8.3%waccr


Determining the Incremental Leverage of a Project• To determine the equity or weighted

average cost of capital for a project, the incremental financing that results if the firm takes on the project needs to be calculated.


Determining the Incremental Leverage of a Project (cont'd)• In other words, what is the change in the

firm’s total debt (net of cash) with the project versus without the project.– Note: The incremental financing of a project

need not correspond to the financing that is directly tied to the project.


Determining the Incremental Leverage of a Project (cont'd)• The following important concepts should

be considered when determining a project’s incremental financing.– Cash Is Negative Debt– A Fixed Payout Policy Implies 100% Debt

Financing– Optimal Leverage Depends on Project and

Firm Characteristics– Safe Cash Flows Can Be 100% Debt Financed


18.6 APV with Other Leverage Policies• Up to this point, it has been assumed the

firm wishes to maintain a constant debt-equity ratio.– Two alternative leverage policies will now be

examined. • Constant interest coverage • Predetermined debt levels


Constant Interest Coverage Ratio

• Constant Interest Coverage Ratio– When a firm keeps its interest payments equal

to a target fraction of its free cash flows• If the target fraction is k, then:

Interest Paid in Year tt k FCF


Constant Interest Coverage Ratio (cont'd)• To implement the APV approach, the

present value of the tax shield under this policy needs to be computed:

– With a constant interest coverage policy, the value of the interest tax shield is proportional to the project’s unlevered value.

(Interest Tax Shield) ( ) ( )

c c

Uc

PV PV k FCF k PV FCF

k V


Constant Interest Coverage Ratio (cont'd)• The value of the levered project, using the

APV method, is:– Levered Value with a Constant Interest

Coverage Ratio

c

c

(interest tax shield)

(1 )

L U U U

U

V V PV V k V

k V


Predetermined Debt Levels

• Rather than set debt according to a target debt-equity ratio or interest coverage level, a firm may adjust its debt according to a fixed schedule that is known in advance.


Predetermined Debt Levels (cont'd)

• Assume now that Avco plans to borrow $30.62 million and then will reduce the debt on a fixed schedule.– $20 million after one year, to $10 million after

two years, and to zero after three years• The RFX project will have no other

consequences for Avco’s leverage.


Table 18.8 Spreadsheet Interest Payments and Interest Tax Shield Given a Fixed Debt Schedule for Avco’s RFX Project



• When debt levels are set according to a fixed schedule, we can discount the predetermined interest tax shields using the debt cost of capital.– For Avco:

2 3

0.73 0.48 0.24(interest tax shield) $1.32 million1.06 1.06 1.06

PV



• The levered value of Avco’s project is:

(Interest tax shield) 59.62 1.32 $60.94 millionL UV V PV



• When a firm has permanent fixed debt, maintaining the same level of debt forever, the levered value of the project simplifies to:– Levered Value with Permanent Debt

• A Cautionary Note:– When debt levels are predetermined, the firm will not

have a target leverage ratio, d. Therefore, equations 18.6, 18.10 and 18.11 are no longer valid. We will need to use more general versions of these equations, provided in section 18.8

L UcV V D


A Comparison of Methods

• Typically, the WACC method is the easiest to use when the firm will maintain a fixed debt-to-value ratio over the life of the investment.

• For alternative leverage policies, the APV method is usually the simplest approach.

• The FTE method is typically used only in complicated settings where the values in the firm’s capital structure or the interest tax shield are difficult to determine.


18.7 Other Effects of Financing

• Issuance and Other Financing Costs– When a firm raises capital by issuing securities,

the banks that provide the loan or underwrite the sale of the securities charge fees.

– These fees should be included as part of the project’s required investment, reducing the NPV of the project.


Table 18.9 Typical Issuance Costs for Different Securities, as a Percentage of Proceeds


Security Mispricing

• If management believes that the securities they are issuing are priced differently than their true value, the NPV of the transaction should be included in the value of the project.– The NPV of the transaction is the difference

between the actual money raised and the true value of the securities sold.


Security Mispricing (cont'd)

• If the financing of the project involves an equity issue, and if management believes that te equity will sell at a price that is less than its true value, this mispricing is a cost of the project for existing shareholders.

• It should be deducted from the project NPV in addition to other issuance costs.


Financial Distress and Agency Costs

• Financial distress and agency costs also impact the cost of capital. – For example, financial distress costs tend to

increase the sensitivity of the firm’s value to market risk, raising the unlevered cost of capital for highly levered firms.


Financial Distress and Agency Costs (cont'd)• The free cash flow estimates for a project

should be adjusted to include expected financial distress and agency costs.

• In addition, because these costs also affect the systematic risk of the cash flows, the unlevered cost of capital will no longer be independent of the firm’s leverage.


18.8 Advanced Topics in Capital Budgeting• Periodically Adjusted Debt

– In the “real world,” most firms allow their debt-equity ratio to stray from their target and periodically adjust leverage to bring it back into line with the target.


Figure 18.1 Firms’ Leverage Policies

Source: J. R. Graham and C. Harvey, “The Theory and Practice of Corporate Finance: Evidence from the Field,” Journal of Financial Economics 60 (2001): 187–243.


18.8 Advanced Topics in Capital Budgeting (cont'd)• Periodically Adjusted Debt

– Suppose the firm adjusts its leverage every s periods, as shown on the next slide. The firm’s interest tax shields up to date s are predetermined and should be discounted at rate rD.


Figure 18.2 Discounting the Tax Shield with Periodic Adjustments



– Interest tax shields that occur after date s depend on future adjustments the firm will make to its debt, so they are risky.

• If the firm will adjust the debt according to a target debt-equity ratio or interest coverage level, then the future interest tax shields should be discounted at rate rD for the periods that they are known, but at rate rU for all earlier periods when they are still risky.



– An important special case is when the debt is adjusted annually

1

1 ( )

(1 ) (1 ) (1 ) 1 c t c t U

c t t tU D U D

Int Int rPV Int

r r r r

1 1

UWACC U c D

D

rr r d r

r

1 1 1

L UUc

D

rV k Vr


Leverage and the Cost of Capital

• When debt is set according to a fixed schedule for some period of time, the interest tax shields for the scheduled debt are known, relatively safe cash flows.– These safe cash flows will reduce the effect of

leverage on the risk of the firm’s equity. – To account for this effect, the value of these

“safe” tax shields should be deducted from the debt, in the same way that we deduct cash, when evaluating a firm’s leverage.


Leverage and the Cost of Capital (cont'd)• If Ts is the present value of the interest tax

shields from predetermined debt, the risk of a firm’s equity will depend on its debt net of the predetermined tax shields:

s sD D T


Leverage and the Cost of Capital (cont'd)• The cost of capital can now be calculated as:

– Leverage and the Cost of Capital with a Fixed Debt Schedule

• The WACC can now be calculated as:– Project WACC with a Fixed Debt Schedule

• Where d is the debt-to-value ratio and Φ = Ts ∕ (cD) is a measure of the permanence of the debt level.

or, equivalently, ( )

s s

U E D E U U Ds s

E D Dr r r r r r rE D E D E

[ ( )wacc U c D U Dr r d r r r


The WACC or FTE Method with Changing Leverage• The WACC and FTE methods are difficult to

use when a firm does not maintain a constant debt-equity ratio. This is because when the proportion of debt financing changes, the project’s equity cost of capital and WACC will not remain constant over time.– However, these methods can still be used with

some adjustments.


Table 18.10 Spreadsheet Adjusted Present Value and Cost of Capital for Avco’s RFX Project with a Fixed Debt Schedule


The WACC or FTE Method with Changing Leverage (cont'd)• For example, at the beginning of the

project, the WACC is calculated as: (1 )

30.32 30.62 9.93% 6% (1 0.40) 6.75%60.94 60.94

wacc E D cE Dr r r

E D E D


The WACC or FTE Method with Changing Leverage (cont'd)• The levered value each year is computed

as:

1 1

1 ( )

LL t tt

wacc

FCF VV

r t


Table 18.11 Spreadsheet WACC Method for Avco’s RFX Project with a Fixed Debt Schedule


Personal Taxes

• The WACC method does not change in the presence of investor taxes.– However, the APV approach requires

modification in the presence of investor taxes because it requires the computation of the unlevered cost of capital.

• rD should be adjusted as:

(1 ) (1 )

iD D

e

r r


Personal Taxes (cont'd)

• The unlevered cost of capital becomes:

• The effective tax rate is:– Unlevered Cost of Capital with Personal Taxes

s

U E Ds s

E Dr r rE D E D

(1 )(1 ) 1 (1 )

c e

i


Personal Taxes (cont'd)

• The interest tax shield is then calculated as:

– The interest tax shields are discounted at rate rU if the firm maintains a target leverage ratio or at rate r *

D if the debt is set according to a predetermined schedule.

* 1Interest Tax Shield in Year D tt r D


Discussion of Data Case Key Topic

• Assume Toyota plans to increase its debt-to-equity ratio by adding $30 billion in debt. How will this increase affect the NPV using each of the three methods?


Chapter Quiz

1. What are the three methods we can use to include the value of the tax shield in the capital budgeting decision?

2. In what situation is the risk of a project likely to match that of the overall firm?

3. Describe the key steps in the WACC valuation method.

4. Describe the adjusted present value (APV) method.


Chapter Quiz (cont’d)5. At what rate should we discount the interest tax

shield in the APV method when a firm maintains a target leverage ratio?

6. Describe the key steps in the flow to equity method for valuing a levered investment.

7. Why does the assumption that the firm maintains a constant debt-equity ratio simplify the flow-to-equity calculation?

8. How do we estimate a project’s unlevered cost of capital when the project’s risk is different from that of a firm?


Chapter Quiz (cont’d)

9. What is the appropriate discount rate for tax shields when the debt schedule is fixed in advance?

10.How would financial distress and agency costs affect a firm’s use of leverage?

11.If the firm’s debt-equity ratio changes over time, can the WACC method still be applied?


18A.1 Deriving the WACC Method

• Consider an investment that is financed by both debt and equity. Because equity holders require an expected return of rE on their investment and debt holders require a return of rD, the firm will have to pay investors:

E DE(1+r )+D(1+r )


18A.1 Deriving the WACC Method (cont’d)• The project generates FCF1 at the end of the year.

In addition, the interest tax shield provides a tax savings of τCrDD. If the investment continues beyond next year, it will have a continuation value of V1

L. So to satisfy investors, the project cash flows must be such that

1 11 1 LE D c DE( +r )+D( +r )=FCF+τ r D+V


18A.1 Deriving the WACC Method (cont’d)

Because V0L=E+D, we can write the WACC

definition in Eq. 18.1 as

If we move the interest tax shield to the left side of the equation, we can use the definition of WACC to rewrite Equation 18A.2:

c 1 11 1 1 LE DE( +r )+D( +r ( - ))=FCF+V

1wacc E D cL L0 0

E Dr = r + r ( -τ )V V

V0L(1+rwacc)



Dividing by (1+rwacc), we can express the value of the investment today as the present value of next period’s free cash flows and continuation value

L 1 10 1

L

wacc

FCF+VV =r



In the same way, we can write the value in one year as the discounted value of the free cash flows and continuation value in year 2. If the WACC is the same the next year, then

2 21

L 1 2 20 2

11 1 (1 )

L

Lwacc

wacc wacc wacc

FCF +VFCF+FCF FCF VrV =

r r r



By repeatedly replacing each continuation value, and assuming the WACC remains constant, we can derive Eq. 18.2:

L 1 2 30 2 3

.....1 (1 ) (1 )wacc wacc wacc

FCF FCF FCFV =r r r


18A.2 The Levered and Unlevered Cost of Capital

Suppose an investor holds a portfolio of all the equity and debt of the firm. Then the investor will receive the free cash flows of the firm plus the tax savings from the interest tax shield. These are the same cash flows an investor would receive from a portfolio of the unlevered firm and a separate “tax shield” security that paid the investor the amount of the tax shield each period.


18A.2 The Levered and Unlevered Cost of Capital (cont’d)

Because the two portfolios generate the same cash flows, by the Law of One Price they have the same market values:

where T is the present value of the interest tax shield. Because these portfolios have equal cash flows, they must also have identical expected returns, which implies:

UE D U TEr +Dr =V r +Tr

L UV =E+D=V +T



Target Leverage RatioSuppose the firm adjusts its debt continuously to maintain a target debt-to-equity ratio, or a target ratio of interest to free cash flows.Because the firm’s debt and interest payments will vary with the firm’s value and cash flows, it is reasonable to expect the risk of the interest tax shield will equal that of the firm’s free cash flows, so rI=rU. Equation 18A.9 becomes

Dividing by (E+D) leads to Eq. 18.6.

U UE D U U U UEr +Dr =V r +Tr =(V +T)r =(E+D)r



Predetermined Debt ScheduleSuppose some of the firm’s debt is set according to a predetermined schedule that is independent of the growth of the firm.suppose the value of the tax shield from the scheduled debt is Ts, and the remaining value of the tax shield T – Ts is from the debt that will be adjusted according to a target leverage ratio. Because the risk of the interest tax shield is similar to the risk of the debt itself, equation 18A.9 becomes

s sT U D

U UE D U UEr +Dr =V r +Tr =V r +(T-T )r +T r



Predetermined Debt Schedule (cont’d)Subtracting TsrD from both sides, and using Ds=D-Ts,

Dividing by (E+Ds) leads to Eq. 18.20.

s sU

U L sE D U

sU

Er +D r =(V +T -T )r (V -T )r(E+D )r


18A.2 The Levered and Unlevered Cost of Capital (cont’d)Risk of the Tax Shield with a Target Leverage Ratio• Under what circumstances is it reasonable to

assume that rT=rU?– We define a target leverage ratio as a setting in which the firm

adjusts its debt at date t to be a proportion of the investment’s value of a proportio of its free cash flow.

– With either policy, the value at date t of the incremental tax shield is proportional to the value of the cash flow, so it should be discounted at the same rate as FCFs.

– Therefore, the assumption follows as long as at each date the cost of capital associated with the value of each future cash flow is the same.


18A.3 Solving for Leverage and Value Simultaneously• When a firm maintains a constant leverage

ratio, to use the APV method we must solve for the debt level and the project value simultaneously.

• The next several slides show how Excel can be used to simultaneously solve for these values.


18A.3 Solving for Leverage and Value Simultaneously• In Table 18A.1, we insert arbitrary values for the

project’s debt capacity. This will not be equal to the required 50% debt-to-value set for the project.

Table 18A.1 Spreadsheet Adjusted Present Value for Avco’s RFX Project with Arbitrary Debt Levels


18A.3 Solving for Leverage and Value Simultaneously (cont’d)• Excel can be instructed to calculate the

spreadsheet iteratively.• Table 18A.2 shows this solution.

Table 18A.2 Spreadsheet Adjusted Present Value for Avco’s RFX Project with Debt Levels Solved Iteratively

berk chapter 18: capital budgeting & leverage

Economy & Finance