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DUKE UNIVERSITYFuqua School of Business

FINANCE 351 - CORPORATE FINANCEProblem Set #4

Prof. Simon Gervais Fall 2011 Term 2

Questions

1. Suppose the corporate tax rate is 40%. Consider a firm that earns $1,000 b efore interestand taxes each year (in perpetuity) with no risk. The firms capital expenditures equal itsdepreciation expenses each year, and it will have no changes to its net working capital. Theriskfree interest rate is 5%.

(a) Suppose the firm has no debt and pays out its net income as a dividend each year. Whatis the value of the firms equity?

(b) Suppose instead the firm makes interest payments of $500 per year. What is the valueof equity? What is the value of debt?

(c) What is the difference between the total value of the firm with leverage and withoutleverage?

(d) The difference in part (c) is equal to what percentage of the value of the debt?

2. Western Lumber Company expects to have a free cash flow of $4.25 million in the comingyear. Free cash flows are expected to grow at a rate of 4% per year thereafter. WesternLumber has an equity cost of capital of 10% and a debt cost of capital of 6%. The corporatetax rate is 35%. If Western Lumber maintains a (constantly rebalanced) debt-equity ratio of0.50, what is the present value of its interest tax shield?

3. Suppose that the Drazil Susej Corporation (DSC) has an equity cost of capital of 8.5%, adebt cost of capital of 7%, a marginal corporate tax rate of 35%, and a debt-equity ratioof 2.6. Suppose that Goodyear does not plan to rebalance its debt, i.e., its current debt isperpetual.

(a) What is DSCs WACC?

(b) What is DSCs unlevered cost of capital?

4. Kurz Manufacturing is currently an all-equity firm with 20 million shares outstanding andstock price of $7.50 per share. Although investors currently expect Kurz to remain an all-equity firm, Kurz plans to announce that it will borrow $50 million and use the funds to

repurchase shares (i.e., Kurzs announcement is not anticipated by investors and thus notreflected in the current stock price). Kurz will pay interest only on this debt, and it has nofurther plans to increase or decrease the amount of debt. Kurz is subject to a 40% corporatetax rate.

(a) What is the market value of Kurzs existing assets before the announcement?

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(b) What is the market value of Kurzs assets (including any tax shields) just after the debtis issued, but before the shares are repurchased?

(c) What is Kurzs share price just before the share repurchase? How many shares will Kurzrepurchase?

(d) What are Kurzs market value balance sheet and share price after the share repurchase?

5. Mercks simplified balance sheets (using book and market values) are currently as follows:

Balance Sheet (book values in millions)

Net working capital 1,473 Long-term debt 5,269Long-term assets 14,935 Equity 11,139

Total assets 16,408 Total liabilities 16,408

Balance Sheet (market values in millions)

Net working capital 1,473 Long-term debt 5,269Market value of long-term assets 51,212 Equity 47,416

Total assets 52,685 Total liabilities 52,685Suppose that Merck decides to move to a 50% b ook debt-to-value ratio by issuing debtand using the proceeds to repurchase shares. The corporate tax rate is 40%; consider onlycorporate taxes. Now construct Mercks balance sheet (with market values only) to reflectthe new capital structure, making sure to add an item called PV(additional tax shields) onthe asset side of the balance sheet. Before it changes its capital structure, Merck has 1,248million shares outstanding. What is the stock price before and after the change?

6. Hula Enterprises is considering a new project to produce solar water heaters. The financemanager wishes to find an appropriate risk adjusted discount rate for the project. The (equity)beta of Hot Water, a firm currently producing solar water heaters, is 1.3. Hot Water has a

debt to total value ratio of 0.4. The expected return on the market is 14% and the risk-freerate is 8%. Throughout this problem, assume that the debt is risk-free and that it is constant(i.e., the debt is never rebalanced).

(a) Suppose the corporate tax rate is 30%. What is the asset (or unlevered) beta for thesolar water heater project?

(b) If Hula is an equity financed firm, what is the weighted average cost of capital for theproject?

(c) If Hula has a debt to equity ratio of 2, what is the weighted average cost of capital forthe project?

(d) Hulas chairman wishes to know why the cost of capital for Hot Water cannot be used

directly. Explain why.(e) The finance manager believes that the solar water heater project can only support

30 cents of debt for every dollar of asset value, i.e., the debt capacity is 30 cents forevery dollar of asset value. This is lower than the 66.67 cents to every dollar of assetvalue (debt to equity ratio of 2) that current projects can support. Current projectshave higher collateral value than the assets of the new solar heater project. Hence she

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is not sure that the debt to equity ratio of 2 used in the weighted average cost of capi-tal calculation is valid. What is the appropriate capital structure to use? What is theweighted average cost of capital that you will arrive at with this capital structure?

7. Company XYZ is currently financed with 40% debt (a 40% debt-to-value ratio). The averagecurrent yield on government securities is 10%. The expected return on the S&P500 portfolio

is 20%. The corporate tax rate is 20% and XYZs stock has a beta of 1.6. XYZ can borrowat 200 basis points above the prevailing government rate.

(a) What is XYZs weighted average cost of capital under the current capital structure?

(b) What is XYZs cost of capital if it decides to pursue a policy of always using 100% equityfinancing instead?

(i) First, solve this under the assumption that XYZs debt will never be rebalanced.

(ii) Now, solve this under the assumption that XYZs debt will be constantlyrebalanced.

8. If it were unlevered, the overall firm beta for Wild Widgets Inc. (WWI) would be 0.9. WWIhas a target debt/equity ratio of 1/2 and plans to constantly rebalance its debt in order

to maintain it. The expected return on the market is 16%, and Treasury bills are currentlyselling to yield 8%. WWI one-year bonds (with a face value of $1,000) carry an annual couponof 7% and are selling for $972.72. The corporate tax rate is 34%.

(a) What is WWIs cost of debt?

(b) What is WWIs weighted average cost of capital?

(c) What is WWIs cost of equity?

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Solutions

1. Because the firms cash flows are riskless, we haverE=rD = 5%.

(a) If the firm has no debt and pays out its net income (of $600 = $1 ,000(1 0.40)) as adividend each year, the firms equity is worth

EU=$600

0.05 = $12,000.

(b) If the firm makes (riskless) interest payments of $500 per year, then the equity-holderswill receive an annual dividend of $300 = ($1,000$500)(10.40). Their equity is thenworth

EL=$300

0.05 = $6,000.

The bondholders receive $500 every year. Their debt is worth

DL =$500

0.05= $10,000.

(c) The levered value of the firm is

VL = EL+ DL= $6,000 + $10,000 = $16,000.

The difference between the levered value and the unlevered value isVLVU= $16,000$12,000 = $4,000.

(d) This difference is the present value of the interest tax shield, which is a fractiontc of thedebtDL:

PV(interest tax shield) = tcDL = (0.40)($10,000) = $4,000.

2. We can calculate the value of Western Lumbers interest tax shield by comparing its valuewith and without leverage. The expected return on Western Lumbers unlevered assets isgiven by

rA=

D

V

rD+

E

V

rE=

0.5

0.5 + 1

(0.06) +

1

0.5 + 1

(0.10) = 8.67%.

If Western Lumber was all-equity financed, then its value would be

VU= $4.25M

0.0867 0.04= $91.07M.

To calculate Western Lumbers levered value, we first need to calculate its weighted-average

cost of capital:

WACC= D

D+ E(1 tc)rD+

E

D+ ErE

= 0.5

0.5 + 1(1 0.35)(0.06) +

1

0.5 + 1(0.10) = 7.97%.

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The levered value of Western Lumber is therefore

VL = $4.25M

0.0797 0.04= $107.14M.

This levered value exceeds the unlevered value by the present value of the interest tax shield,

that is, by

PV(interest tax shield) =VL VU= $107.14M $91.07M = $16.07M.

3. We have rE = 8.5%, rD = 7%, tc = 0.35, and D/E = 2.6 (which corresponds to D/V =2.6

2.6+1= 0.722).

(a) DSCs weighted average cost of capital is

WACC=D

V(1 tc)rD+

E

VrE= (0.722)(1 0.35)(0.07) + (0.278)(0.085) = 5.65%.

(b) We know that the weighted average cost of capital satisfies

WACC=

1 tc

D

V

rA 5.65% =

1 (0.35)(0.722)

rA.

This implies that rA= 7.56%.

4. (a) Because Kurz Manufacturing is initially all-equity financed, its value is the value of itsequity:

VU=EU= 20 million $7.50 = $150 million.

(b) Right after the debt is issued, Kurzs value increases by the amount of debt it raised($50 million), and by the tax shield that this debt creates (0 .40$50 million = $20 million).

That is, its value increases by $70 million.(c) Before the debt is repurchased, the total value of the firm is $150 million + $70 million =

$220 million. Since the debt is worth $50 million (i.e., the debtholders get what they payfor), the equity must be worth $220 million $50 million = $170 million. This impliesthat Kurzs share price is $170 million 20 million = $8.50. This in turn implies thatthe $50 million raised through the debt issue will allow Kurz to buy back $50 million $8.50 = 5.882 million shares.

(d) After the share repurchase, the total market value of Kurz Manufacturing is $220 million$50 million = $170 million. On the asset side of the balance sheet, Kurzs unlevered as-sets are worth $150 million, as before, and the debt tax shield is $20 million. On theliability side of the balance sheet, the debt is worth $50 million, and the equity is worth

$120 million (20 million 5.882 million = 14.118 million shares trading at $8.50 each).

5. The long-term debt is increased from $5,269 to

50% of book = 50% $16,408 = $8,204,

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that is it is increased by $8,204 $5,269 = $2,935. This implies that the debt tax shield isincreased by 0.40$2,935 = $1,174, and the market value of the firm is now $52,685+$1,174 =$53,859. The new balance sheet (with market values) will look as follows:

Balance Sheet (market values in millions)

Net working capital 1,473 Long-term debt 8,204Mkt value of long-term assets 51,212 Equity 45,655PV(additional tax shields) 1,174

Total assets 53,859 Total liabilities 53,859

Before the change, the equity is worth $47,416, and 1,248 shares are outstanding, so that thestock price is

P =47,416

1,248 = 37.99.

Upon the firms announcement of its plans to repurchase shares, the firms value shouldgo up by the extra debt tax shield that this will generate. That is, the equity goes up to$47,416 + $1,174 = $48,590, and each share is then worth

P =48,5901,248

= 38.93.

The amount raised from the new debt, $2,935, is therefore used to repurchase

2,935

38.93= 75.38 shares.

6. (a) When the tax rate is 30%, the asset beta (unlevered beta) of the solar heater project is

A= E

1 + (1 tc)DE

= 1.3

1 + (1 0.30)23

= 0.8863.

Notice that the debt-to-equity ratio is 0.4

0.6 = 2

3 when the debt-to-value ratio is 0.4.(b) If Hula is an all-equity financed firm, the weighted average cost of capital for the project

is just the unlevered cost of equity. The unlevered cost of equity is given by the CAPM:

rA= runlevered

E =rf+ A(rm rf) = 0.08 + (0.8863)(0.14 0.08) = 13.32%.

(c) The levered beta corresponding to a debt-to-equity ratio of 2 is

leveredE =

1 + (1 tc)

D

E

A= [1 + (1 0.30)(2)] (0.8863) = 2.13.

Hence the return on equity corresponding to the leverage of 2 is given by the CAPM:

rleveredE

= 0.08 + (2.13)(0.14 0.08) = 20.8%.

Thus the weighted average cost of capital for the project is

WACC=

D

V

p(1 tc)r

pD+

E

V

prpE

= 0.667(1 0.30)(0.08) + 0.333(0.208) = 10.7%.

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(d) Hot Water may have a different capital structure and thus its weighted average cost ofcapital is not applicable. Even though the business risk is the same, the difference incapital structure implies a different weighted average cost of capital.

(e) You are explicitly told that the new project has a lower debt capacity. Thus it can onlysupport 30 cents of debt for every dollar of asset value. Existing assets can support

66.67 cents. Thus the long run debt capacity of the new project is different. Hence weneed to account for this while doing our capital budgeting. The appropriate leverageadjustment is the debt to value ratio of 0.3. First, the levered beta corresponding to thedebt to total value ratio of 0.3 is calculated as follows:

leveredE =

1 + (1 tc)

D

E

A=

1 + (1 0.30)

0.3

0.7

(0.8863) = 1.15.

The corresponding return on equity, as given by the CAPM, is

rE= 0.08 + 1.15(0.14 0.08) = 14.9%,

and the weighted average cost of capital is then

WACC=

D

V

p(1 tc)r

pD+

E

V

prpE

= 0.3(1 0.30)(0.08) + 0.7(0.149) = 12.1%.

Hence 12.1% rather than 10.7% is the right answer.

7. (a) With a debt-to-value ratio of 40%, we have DE = 40%60%

= 2/3. We use the CAPM tocompute the cost of equity:

rE=rf+ E(rm rf) = 0.10 + 1.6(0.20 0.10) = 26%.

Noticing that rD = 10% + 2% = 12%, we have

WACC=

D

V

(1 tc)rD+

E

V

rE= 0.40(1 0.20)(0.12) + 0.60(0.26) = 19.44%.

(b) The risk premium on XYZs debt is 0.02 =D(0.20 0.10), for a D of 0.2.

(i) When the debt is permanent, we can unlever the equity beta to get an (unlevered)asset beta as follows:

A=E+

DE

(1 tc)D

1 +DE

(1 tc)=

1.6 + 23

(1 0.20)(0.2)

1 + 23

(1 0.20)= 1.113.

Using CAPM, we find

rA=rf+ A(rm rf) = 0.10 + 1.113(0.20 0.10) = 21.13%.

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(ii) When the debt is constantly rebalanced, we can unlever the equity beta to get an(unlevered) asset beta as follows:

A=

D

V

D+

E

V

E= (0.40)(0.2) + (0.60)(1.6) = 1.040.

Using CAPM, we find

rA=rf+ A(rm rf) = 0.10 + 1.040(0.20 0.10) = 20.40%.

8. From the data we have rm = 0.16, rf = 0.08, D/V = 1/3, and E/V = 2/3. Also the beta ofan otherwise identical but unlevered firm is 0.9. This means that

rA= rf+ (rm rf)A= 0.08 + (0.16 0.08)(0.9) = 15.2%.

(a) WWI one-year coupon bonds have a face value of $1,000. One year from now, they willpay the face value and the 7% coupon, i.e. 1,000(1 + 0.07) = 1,070. Since WWIs bondsare now selling for $972.72, we can deduct the value for WWIs (pre-tax) cost of debt:

rD = 1,070

972.72 1 = 10%.

(b) The weighted average cost of capital for a levered firm can b e calculated as follows:

WACCL=rA D

VtcrD = 0.152

1

3(0.34)(0.10) = 14.06667%.

(c) Since

WACCL =rD(1 tc)D

V + rE

E

V,

we have0.1406667 = 0.10(1 0.34)

1

3+ rE

2

3,

which implies rE= 17.80%.

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