18: Chapter Valuation and Capital Budgeting for the Levered Firm
18.1 a. The maximum price that Hertz should be willing to pay for the fleet of cars with allequity 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 + (1-0.40)($300,000)A50.10 + PVCCATS
InvestmentxTaxRatexCCA 1 0.50( DiscountRate) ][ ] CCA DiscountRate 1 DiscountRate Px 0.40 x 0.25 1 0.50(0.10) [ ][ ] 0.2727 P 0.25 0.10 1 0.10 PVCCATS [
Set the NPV equal to zero. 0 = -P + (1-0.40)($300,000)A50.10 + 0.2727P 0.7273P= $682,341.62 P= $938,184.55 Therefore, the most that Hertz should be willing to pay for the fleet of cars with allequity funding is $938,184.55. 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 Hertzs case, the NPV of financing side effects equals the after-tax present value of the cash flows resulting from the firms 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) Hertz paid $975,000 for the fleet of cars.
InvestmentxTaxRatexCCA 1 0.50( DiscountRate) ][ ] CCA DiscountRate 1 DiscountRate 975,000 x 0.40 x 0.25 1 0.50(0.10) [ ][ ] $265,909.09 0.25 0.10 1 0.10 PVCCATS [
NPV = -$975,000 + (1- 0.4)($300,000)A50.10 + 265,909.09 = -975,000 + 682,341.62+265,909.09 = -$26,749.29
Answers to End-of-Chapter Problems B-23
NPV(Financing Side Effects) The net present value of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. NPV(Financing Side Effects) PV(Principal Payments) = Proceeds After-Tax PV(Interest Payments)
Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 8%. NPV(Financing Side Effects) = $600,000 (1 0.40)(0.08)($600,000)A50.08 [$600,000/(1.08)5] = $600,000 114,990.05 408,349.92 = 76,660.03 APV APV = NPV(All-Equity) + NPV(Financing Side Effects) = -$26,749.29+ $76,660.03 = $ 49,910.74 Therefore, if Hertz uses $600,000 of five-year, 8% debt to fund the $975,000 purchase, the Adjusted Present Value (APV) of the project is $ 49,910.74. c. To determine the maximum price, set the APV=0 = NPV (All equity) + NPV(Loan) 0 = -P + (1-0.40)($300,000)A50.10 + 0.2727P + $600,000 (1 0.40)(0.05)($600,000)A50.08 [$600,000/(1.08)5] 0 = -0.7273P +682,341.62 + 600,000 71,868.78 408,350 0.7273P = 802,122.84 P = 1,102,877.55 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. In Peatcos case, the NPV of financing side effects equals the after-tax present value of the cash flows resulting from the firms debt. APV = NPV(All-Equity) + NPV(Financing Side Effects) NPV(All-Equity) NPV = -Initial Investment + PV[(1-TC)(Earnings Before Taxes and Depreciation)] + PV(CCA Tax Shield) Assuming the company has other assets in this CCA pool,
18.2
a.
InvestmentxTaxRatexCCA 1 0.50( DiscountRate) ][ ] CCA DiscountRate 1 DiscountRate 2,100, 000 x0.40 x0.30 1 0.50(0.16) [ ][ ] $510, 045 0.30 0.16 1 0.16 PVCCATS [
B-24
Answers to End-of-Chapter Problems
NPV = -$2,100,000 + (1-0.40)($900,000)A30.16 + $510,045 = -$377,175 NPV(Financing Side Effects) The net present value of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. NPV(Financing Side Effects) = -Flotation costs+ NPV(tax shield on flotation costs)+Amount borrowed - PV (Tax shield on interest) Present Value of loan payments Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 7.5%. Since $21,000 in flotation costs will be amortized over the three-year life of the loan, $7,000 = ($21,000 / 3) of flotation costs will be expensed per year.
NPV of Flotation costs = -21,000 + (0.40)($7,000)A30.075 = - 21,000 +7,281= -$13,719 NPV(Loan) = ($2,100,000 (1 0.40)(0.075)($2,100,000)A30. 075
[$2,100,000/(1.075)3] + = $2,100,000 - $245,750-$1,690,417 = $163,833 APV APV = NPV(All-Equity) + NPV(Flotation costs) + NPV(Loan) = -$377,175 - $13,719 +163,833 = -$227,061 Since the adjusted present value (APV) of the project is negative, Peatco should not undertake the project. b. NPV(Financing Side Effects) The only change is that the interest cost will be based on 6.5% but the cash flows will still be discounted at the companys cost of debt which is 7.5%. The net present value of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. NPV(Financing Side Effects) = -Flotation costs+ NPV(tax shield on flotation costs)+Amount borrowed - PV (Tax shield on interest) Present Value of loan payments Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 7.5%. Since $21,000 in flotation costs will be amortized over the three-year life of the loan, $7,000 = ($21,000 / 3) of flotation costs will be expensed per year.
NPV of Flotation costs = -21,000 + (0.40)($7,000)A30.075
Answers to End-of-Chapter Problems B-25
= - 21,000 +7,281= -$13,719 NPV(Loan) = ($2,100,000 (1 0.40)(0.065)($2,100,000)A30. 075
[$2,100,000/(1.075)3] + = $2,100,000 - $212,983-$1,690,417 = $196,600 APV APV = NPV(All-Equity) + NPV(Flotation costs) + NPV(Loan) = -$377,175 - $13,719 +196,600 = -$194,294 Since the adjusted present value (APV) of the project is still negative, Peatco should still not undertake the project. Note: When a subsidy exists on a loan, you cannot use the PV(Tax Shield) method to compute the NPV of the loan. This is true because even with an interest rate subsidy, the appropriate rate by which to discount the after-tax interest payments is the market rate of interest. The PV(Tax Shield) method presumes the interest rate and the discount rate are the same. 18.3 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. According to Modigliani-Miller Proposition II with corporate taxes: rS = r0 + (B/S)(r0 rB)(1 TC) where r0 = the required return on the equity of an unlevered firm rS = the required return on the equity of a levered firm rB = the pre-tax cost of debt TC = the corporate tax rate B/S = the firms debt-to-equity ratio
In this problem: rS = 0.18 rB = 0.10 TC = 0.40 B/S = 0.25 Solve for MVPs unlevered cost of capital (r0): rS = r0 + (B/S)(r0 rb)(1 TC) 0.18 = r0 + (0.25)(r0 0.10)(1 0.40) r0 = 0.17 The cost of MVPs unlevered equity is 17%.
Answers to End-of-Chapter Problems B-26
APV = NPV(All-Equity) + NPV(Financing Side Effects) NPV(All-Equity) NPV = PV(Unlevered Cash Flows) = -$15,000,000 + $4,000,000/1.17 + $8,000,000/(1.17)2 + $9,000,000/(1.17)3 = -$117,753 NPV(Financing Side Effects) The net present value of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. NPV(Financing Side Effects) = Proceeds After-Tax PV(Interest Payments) PV(Principal Payments)
Year Outstanding Debt at the Start of the Year (B) Debt Repayment at the End of the Year 1 $6,000,000 $2,000,000 2 $4,000,000 $2,000,000 3 $2,000,000 $2,000,000 4+ $0 $0
Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 10%. NPV(Financing Side Effects) = $6,000,000 (1 0.40)(0.10)($6,000,000) / (1.10) $2,000,000/(1.10) (1 0.40)(0.10)($4,000,000)/(1.10)2 $2,000,000/(1.10)2 (1 0.40)(0.10)($2,000,000)/(1.10)3 $2,000,000/(1.10)3 = $410,518 APV APV = NPV(All-Equity) + NPV(Financing Side Effects) = -$117,753 + $410,518 = $292,765 Since the adjusted present value (APV) of the project is positive, MVP should proceed with the expansion. 18.4 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 ventures case, the NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. APV = NPV(All-Equity) + NPV(Financing Side Effects) NPV(All-Equity) NPV = -Initial Investment + PV[(1 TC)(Earnings Before Interest, Taxes, and Depreciation )] + PV(CCA Tax Shield)
Answers to End-of-Chapter Problems B-27
Assuming that the company has other assets in the class,
InvestmentxTaxRatexCCA 1 0.50( DiscountRate) ][ ] CCA DiscountRate 1 DiscountRate 20, 000, 000 x0.25 x0.30 1 0.50(0.12) [ ][ ] $3,380,102 0.30 0.12 1 0.12 PVCCATS [
NPV = -$20,000,000 + [(1-0.25)($3,000,000)A200.12] + 3,380,102 = -$20,000,000 + $16,806,248 + 3,380,102 = $186,350 NPV(Financing Side Effects) The NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 10%. NPV(Financing Side Effects) = Proceeds After-tax PV(Interest Payments) PV(Principal Repayments) = $10,000,000 (1 0.25)(0.05)($10,000,000)A150.09 [$10,000,000/((1.09)15] = $4,231,861 APV APV = NPV(All-Equity) + NPV(Financing Side Effects) = $186,350 + $4,231,861 = $4,418,211 The Adjusted Present Value (APV) of the project is $4,418,211. 18.5 a. In order to value a firms equity using the Flow-to-Equity approach, discount the cash flows available to equity holders at the cost of the firms levered equity (rS).
Sales Cost of Goods Sold General and Administrative Costs Interest Expense Pre-Tax Income Taxes at 40% Cash Flow Available to Equity Holders One Restaurant Milano Pizza Club $1,000,000 $3,000,000 ($400,000) ($1,200,000) ($300,000) ($900,000) ($25,650) ($76,950) $274,350 $823,050 ($109,740) ($329,220) $164,610 $493,830
Since this cash flow will remain the same forever, the present value of cash flows available to the firms equity holders is a perpetuity of $493,830, discounted at 21%. PV(Flows-to-Equity)
Answers to End-of-Chapter Problems
= $493,830 / 0.21
B-28
= $2,351,571 The value of Milano Pizza Clubs equity is $2,351,571. b. The value of a firm is equal to the sum of the market values of its debt and equity. VL = B + S The market value of Milano Pizza Clubs equity (S) is $2,351,571 (see part a). The problem states that the firm has a debt-to-equity ratio of 30%, which can be written algebraically as: B / S = 0.30 Since S = $2,351,571: B / $2,351,571 = 0.30 B = $705,471 The market value of Milano Pizza Clubs debt is $705,471, and the value of the firm is $3,057,042 (= $705,471 + $2,351,571). The value of Milano Pizza Club is $3,057,042. 18.6 a. In order to determine the cost of the firms debt (rB), solve for the discount rate that makes the present value of the bonds future cash flows equal to the bonds current price. Since WWIs one-year, $1,000 par value bonds carry a 7% coupon, bond holders will receive a payment of $1,070 =[$1,000 + (0.07)($1,000)] in one year. $972.73 = $1,070/ (1+ rB) rB = 0.10 Therefore, the cost of WWIs debt is 10%. b. Use the Capital Asset Pricing Model to find the return on WWIs unlevered equity (r0). According to the Capital Asset Pricing Model: r0 = rf + Unlevered(rm rf) where r0 = the cost of a firms unlevered equity rf = the risk-free rate rm= the expected return on the market portfolio Unlevered = the firms beta under all-equity financing In this problem: rf = 0.08
B-29
Answers to End-of-Chapter Problems
rm= 0.16 Unlevered = 0.9 r0 = rf + Unlevered(rm rf) = 0.08 + 0.9(0.16-0.08) = 0.152 The cost of WWIs unlevered equity is 15.2%. Next, find the cost of WWIs levered equity. According to Modigliani-Miller Proposition II with corporate taxes rS = r0 + (B/S)(r0 rB)(1 TC) where r0 = the cost of a firms unlevered equity rS = the cost of a firms levered equity rB = the pre-tax cost of debt TC = the corporate tax rate B/S = the firms target debt-to-equity ratio
In this problem: r0 = 0.152 rB = 0.10 TC = 0.34 B/S = 0.50 The cost of WWIs levered equity is: rS = r0 + (B/S)(r0 rB)(1 TC) = 0.152 + (0.50)(0.152-0.10)(1 0.34) = 0.1692 The cost of WWIs levered equity is 16.92%. c. In a world with corporate taxes, a firms weighted average cost of capital (rwacc) is equal to: rwacc where = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS B / (B+S) = the firms debt-to-value ratio S / (B+S) = the firms equity-to-value ratio rB = the pre-tax cost of debt rS = the cost of equity TC = the corporate tax rate
The problem does not provide either WWIs debt-to-value ratio or WWIs equity-to-value ratio. However, the firms debt-to-equity ratio of 0.50 is given, which can be written algebraically as:
Answers to End-of-Chapter Problems B-30
B / S = 0.50 Solving for B: B = (0.5 * S) A firms debt-to-value ratio is: Since B = (0.5 * S): WWIs debt-to-value ratio = (0.5 * S) / { (0.5 * S) + S} = (0.5 * S) / (1.5 * S) = 0.5 / 1.5 = 1/3 WWIs debt-to-value ratio is 1/3. A firms equity-to-value ratio is: S / (B+S) Since B = (0.5 * S): WWIs equity-to-value ratio = S / {(0.5*S) + S} = S / (1.5 * S) = (1 / 1.5) = 2/3 B / (B+S)
WWIs equity-to-value ratio is 2/3. Thus, in this problem: B / (B+S) S / (B+S) rB rS TC rwacc = 1/3 = 2/3 = 0.10 = 0.1692 = 0.34 = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS = (1/3)(1 0.34)(0.10) + (2/3)(0.1692) = 0.1348
WWIs weighted average cost of capital is 13.48%. 18.7 a. Bolero has a capital structure with three parts: long-term debt, short-term debt, and equity. i. Book Value Weights:
Type of Financing Long-term debt Short-term debt Common Stock Total Answers to End-of-Chapter Problems Book Value $5,000,000 $5,000,000 $10,000,000 $20,000,000 Weight 25% 25% 50% 100% Cost 10% 8% 15%
B-31
Since interest payments on both long-term and short-term debt are tax-deductible, multiply the pre-tax costs by (1-TC) to determine the after-tax costs to be used in the weighted average cost of capital calculation. rwacc = (WeightLTD)(CostLTD)(1-TC) + (WeightSTD)(CostSTD)(1-TC) + (WeightEquity)(CostEquity) = (0.25)(0.10)(1-0.4) + (0.25)(0.08)(1-0.4) + (0.50)(0.15) = 0.102 If Bolero uses book value weights, the firms weighted average cost of capital would be 10.2%. ii. Market Value Weights: Type of Market Financing Value Weight Cost Long-term debt $2,000,000 10% 10% Short-term debt $5,000,000 25% 8% Common Stock $13,000,000 65% 15% Total $20,000,000 100% Since interest payments on both long-term and short-term debt are tax-deductible, multiply the pre-tax costs by (1-TC) to determine the after-tax costs to be used in the weighted average cost of capital calculation. rwacc = (WeightLTD)(CostLTD)(1-TC) + (WeightSTD)(CostSTD)(1-TC) + (WeightEquity)(CostEquity) = (0.10)(0.10)(1-0.4) + (0.25)(0.08)(1-0.4) + (0.65)(0.15) = 0.1155 If Bolero uses market value weights, the firms weighted average cost of capital would be 11.55%. iii. Target Weights: If Bolero has a target debt-to-equity ratio of 100%, then both the target equity-to-value and target debt-to-value ratios must be 50%. Since the target values of long-term and short-term debt are equal, the 50% of the capital structure targeted for debt would be split evenly between long-term and short-term debt (25% each).
Type of Financing Target Weight Long-term debt 25% Short-term debt 25% Common Stock 50% Total 100% Cost 10% 8% 15%
Since interest payments on both long-term and short-term debt are tax-deductible, multiply the pre-tax costs by (1-TC) to determine the after-tax costs to be used in the weighted average cost of capital calculation. rwacc = (WeightLTD)(CostLTD)(1-TC) + (WeightSTD)(CostSTD)(1-TC) + (WeightEquity)(CostEquity) = (0.25)(0.10)(1-0.4) + (0.25)(0.08)(1-0.4) + (0.50)(0.15) = 0.102
Answers to End-of-Chapter Problems
B-32
If Bolero uses target weights, the firms weighted average cost of capital would be 10.2%. b. The differences in the WACCs are due to the different weighting schemes. The firms 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. In a world with corporate taxes, a firms weighted average cost of capital (rwacc) equals: rwacc where = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS B / (B+S) S / (B+S) rB rS TC = the firms debt-to-value ratio = the firms equity-to-value ratio = the pre-tax cost of debt = the cost of equity = the corporate tax rate
18.8
a.
The market value of Neons debt is $24 million, and the market value of the firms equity is $60 million (= 4 million shares * $15 per share). Therefore, Neons current debt-to-value ratio is 28.57% [= $24 / ($24 + $60)], and the firms current equity-to-value ratio is 71.43% [= $60 / ($24 + $60)]. Since Neons CEO believes its current capital structure is optimal, these values can be used as the target weights in the firms weighted average cost of capital calculation. Neons bonds yield 11% per annum. Since the yield on a firms bonds is equal to its pretax cost of debt, rB equals 11%. Use the Capital Asset Pricing Model to determine Neons cost of equity. According to the Capital Asset Pricing Model: rS = rf + Equity(rm rf) where rS = the cost of a firms equity rf = the risk-free rate rm- rf = the expected market risk premium Equity = the firms equity beta Equity = [Covariance(Stock Returns, Market Returns)] / Variance(Market Returns) The covariance between Neons stock returns and returns on the market portfolio is 0.031. The standard deviation of market returns is 0.16. The variance of returns is equal to the standard deviation of those returns squared. The variance of the returns on the market portfolio is 0.0256 [= (0.16)2]. Neons equity beta is 1.21 (= 0.031 / 0.0256).
Answers to End-of-Chapter Problems B-33
The inputs to the CAPM in this problem are: rf = 0.07 rm- rf = 0.085 Equity = 1.21 rS = rf + Equity(rm rf) = 0.07 + 1.21(0.085) = 0.1729 The cost of Neons equity (rS) is 17.29%. The inputs for the weighted average cost of capital calculation are: B / (B+S) S / (B+S) rB rS TC rwacc = 0.2857 = 0.7143 = 0.11 = 0.1729 = 0.34
= {B / (B+S)}(1 TC) rB + {S / (B+S)}rS = (0.2857)(1 0.34)(0.11) + (0.7143)(0.1729) = 0.1442
Neons weighted average cost of capital is 14.42%, Use the weighted average cost of capital to discount Neons expected unlevered cash flows. NPV = -$27,500,000 + $9,000,000A50.1442 = $3,088,379 Since the NPV of the equipment is positive, Neon should make the purchase. 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. It will remain 14.42%. The weighted average cost of capital is based on target capital structure weights. Since the current capital structure is optimal, all-debt funding for the project simply implies that the firm will have to use more equity in the future to bring the capital structure back towards the target. In a world with corporate taxes, a firms weighted average cost of capital (rwacc) equals: rwacc where = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS B / (B+S) = the firms debt-to-value ratio S / (B+S) = the firms equity-to-value ratio rB = the pre-tax cost of debt rS = the cost of equity TC = the corporate tax rate
B-34
18.9
a.
Answers to End-of-Chapter Problems
Since the firms target debt-to-equity ratio is 200%, the firms target debt-to-value ratio is 2/3, and the firms target equity-to-value ratio is 1/3. The inputs to the WACC calculation in this problem are: B / (B+S) S / (B+S) rB rS TC rwacc = 2/3 = 1/3 = 0.10 = 0.20 = 0.34 = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS = (2/3)(1 0.34)(0.10) + (1/3)(0.20) = 0.1107
NECs weighted average cost of capital is 11.07%. Use the weighted average cost of capital to discount NECs unlevered cash flows. NPV = -$20,000,000 + $8,000,000 / 0.1107 = $52,267,389 Since the NPV of the project is positive, NEC should proceed with the expansion. 18.10 a. ABC was an all-equity firm prior to its recapitalization. The value of ABC as an allequity firm equals the present value of after-tax cash flows, discounted at the cost of the firms unlevered equity of 18%. VU = [(Pre-Tax Earnings)(1 TC)] / r0 = [($30,000,000)(1 0.34)] / 0.18 = $110,000,000 The value of ABC before the recapitalization is announced is $110 million. Since ABC is an all-equity firm, the value of ABCs equity before the announcement is also $110 million. ABC has 1 million shares of common stock outstanding. The price per share before the announcement is $110 (= $110 million / 1 million shares) b. The adjusted present value of a firm equals it value under all-equity financing (VU) plus the net present value of any financing side effects. In ABCs case, the NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. APV = VU + NPV(Financing Side Effects) From part a: VU = $110,000,000 NPV(Financing Side Effects)
Answers to End-of-Chapter Problems B-35
The NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 10%. NPV(Financing Side Effects) = Proceeds After-tax PV(Interest Payments) = $50,000,000 (1 0.34)(0.10)($50,000,000)/0.10 = $17,000,000 APV APV = VU + NPV(Financing Side Effects) = $110,000,000 + $17,000,000 = $127,000,000 The value of ABC after the recapitalization plan is announced is $127 million. Since ABC has not yet issued the debt, the value of ABCs equity after the announcement is also $127 million. ABC has 1 million shares of common stock outstanding. The price per share after the announcement is $127 (= $127 million / 1 million shares). c. ABC will receive $50 million in cash as a result of the debt issue. Since the firms stock is worth $127 per share, ABC will repurchase 393,701 shares (= / $50,000,000 $127 per share). After the repurchase, the firm will have 606,299 (= 1,000,000 393,701) shares of common stock outstanding. Since the value of ABC is $127 million and the firm has $50 million of debt, the value of ABCs equity after the recapitalization is $77 million (= $127 million - $50 million). ABC has 606,299 shares of common stock outstanding after the recapitalization. The price per share after the repurchase is $127 (= $77 million / 606,299 shares). d. In order to value a firms equity using the Flow-to-Equity approach, discount the cash flows available to equity holders at the cost of the firms levered equity (rS). According to Modigliani-Miller Proposition II with corporate taxes rS = r0 + (B/S)(r0 rB)(1 TC) where r0 rS rB TC B S = the required return on the equity of an unlevered firm = the required return on the equity of a levered firm = the pre-tax cost of debt = the corporate tax rate = the market value of the firms debt = the market value of the firms equity
In this problem:
Answers to End-of-Chapter Problems B-36
r0 rB TC B S
= 0.18 = 0.10 = 0.34 = $50,000,000 = $77,000,000
The required return on ABCs levered equity after the recapitalization is: rS = r0 + (B/S)(r0 rB)(1 TC) = 0.18 + ($50,000,000 / $77,000,000)(0.18 0.10)(1 0.34) = 0.2143 The required return on ABCs levered equity after the recapitalization is 21.43%. Since ABC has $50,000,000 of 10% debt, the firm will make interest payments of $5,000,000
Cash Flows to Equity EBIT $30,000,000 Interest 5,000,000 Pre-Tax Earnings 25,000,000 Taxes at 34% 8,500,000 After-Tax Earnings 16,500,000
Interest (= $50,000,000 * 0.10) at the end of each year. Since the firm pays all of its after-tax earnings out as dividends at the end of each year, equity holders will receive $16,500,000 of cash flow per year in perpetuity. S = Cash Flows Available to Equity Holders / rS = $16,500,000 / 0.2143 = $77 million Note: the unrounded cost of equity of 21.42857143% must be used to calculate the exact answer. The value of ABCs equity after the recapitalization is $77 million. 18.11 a. If Mojito 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 of 16%.
Mojito Mint Company Sales Revenue Variable Costs EBIT Taxes at 40% Unlevered After-Tax Earnings $19,740,000 11,844,000 7,896,000 3,158,400 4,737,600
Answers to End-of-Chapter Problems
B-37
VU = $4,737,600 / 0.16 = $29,610,000 Therefore, Mojito Mint Company would be worth $29,610,000 as an unlevered firm. b. According to Modigliani-Miller Proposition II with corporate taxes: rS = r0 + (B/S)(r0 rB)(1 TC) where r0 = the required return on the equity of an unlevered firm rS = the required return on the equity of a levered firm rB = the pre-tax cost of debt TC = the corporate tax rate B/S = the firms debt-to-equity ratio In this problem: r0 = 0.16 rB = 0.10 TC = 0.40 B/S = 2/3 The required return on Mojitos levered equity is: rS = r0 + (B/S)(r0 rB)(1 TC) = 0.16 + (2/3)(0.16 0.10)(1 0.40) = 0.184 The required return on Mojitos levered equity (rS) is 18.4%. c. In a world with corporate taxes, a firms weighted average cost of capital (rwacc) equals: rwacc where = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS B / (B+S) S / (B+S) rB rS TC = the firms debt-to-value ratio = the firms equity-to-value ratio = the pre-tax cost of debt = the cost of equity = the corporate tax rate
The problem does not provide either Mojitos debt-to-value ratio or Mojitos equity-tovalue ratio. However, the firms debt-to-equity ratio of 2/3 is given, which can be written algebraically as: B / S = 2/3 Solving for B: B = (2/3)(S) A firms debt-to-value ratio is:
Answers to End-of-Chapter Problems
B / (B+S)
B-38
Since B = (2/3)(S): Mojitos debt-to-value ratio = (2/3)(S) / { (2/3)(S) + S} = (2/3)(S) / (5/3)(S) = (2/3)/(5/3) = 2/5 Mojitos debt-to-value ratio is 2/5. A firms equity-to-value ratio is: S / (B+S) Since B = (2/3)(S): Mojitos equity-to-value ratio = S / {(2/3)(S) + S} = S / (5/3)(S) = (1 / (5/3)) = 3/5
Mojitos equity-to-value ratio is 3/5. The inputs to the WACC calculation are: B / (B+S) S / (B+S) rB rS TC rwacc = 2/5 = 3/5 = 0.10 = 0.184 = 0.40 = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS = (2/5)(1 0.40)(0.10) + (3/5)(0.184) = 0.1344
Mojitos weighted average cost of capital is 13.44%. Use the weighted average cost of capital to discount the firms unlevered after-tax earnings. VL = $4,737,600 / 0.1344 = $35,250,000 Therefore, the value of Mojito Mint Company is $35,250,000. Since the firms equity-to-value ratio is 3/5, the value of Mojitos equity is $21,150,000 {= (3/5)($35,250,000)}. Since the firms debt-to-value ratio is 2/5, the value of Mojitos debt is $14,100,000 {= (2/5)( $35,250,000)}.
Answers to End-of-Chapter Problems
B-39
d.
In order to value a firms equity using the Flow-to-Equity approach, discount the cash flows available to equity holders at the cost of the firms levered equity (rS). Since the pre-tax cost of the firms debt is 10%, and the firm has $14,100,000 of debt outstanding, Mojito must pay $1,410,000 (= 0.10 * $14,100,000) in interest at the end of each year.
Mojito Mint Company Sales Revenue Variable Costs EBIT Interest Pre-Tax Earnings Taxes at 40% After-Tax Earnings 19,740,000 11,844,000 7,896,000 1,410,000 6,486,000 2,594,400 3,891,600
Since the firm pays all of its after-tax earnings out as dividends at the end of each year, equity holders will receive $3,891,600 of cash flow per year in perpetuity. S = Cash Flows Available to Equity Holders / rS = $3,891,600 / 0.184 = $21,150,000 The value of Mojitos equity is $21,150,000. 18.12 a. Since Lone Star 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 of 20%.
Lone Star Industries EBIT Taxes at 40% Unlevered After-Tax Earnings $152.00 $60.80 $91.20
VU = $91.20/ 0.20 = $456 Lone Star Industries is worth $456 as an unlevered firm. b. The adjusted present value of a firm equals its value under all-equity financing (VU) plus the net present value of any financing side effects. In ABCs case, the NPV of financing side effects equals the after-tax present value of cash flows resulting from debt. APV = VU + NPV(Financing Side Effects) From part a: VU = $456 NPV(Financing Side Effects)
Answers to End-of-Chapter Problems B-40
The NPV of financing side effects equals the after-tax present value of cash flows resulting from the firms debt. Given a known level of debt, debt cash flows should be discounted at the pre-tax cost of debt (rB), 10%. NPV(Financing Side Effects) = Proceeds After-tax PV(Interest Payments) = $500 (1 0.40)(0.10)($500)/0.10 = $200 APV APV = VU + NPV(Financing Side Effects) = $456 + $200 = $656 The value of Lone Star Industries is $656 with leverage. Since Lone Star has $500 of debt, the value of the firms equity is $156 = ($656 - $500). c. According to Modigliani-Miller Proposition II with corporate taxes rS = r0 + (B/S)(r0 rB)(1 TC) where r0 rS rB TC B S = the required return on the equity of an unlevered firm = the required return on the equity of a levered firm = the pre-tax cost of debt = the corporate tax rate = the market value of the firms debt = the market value of the firms equity
In this problem: r0 rB TC B S = 0.20 = 0.10 = 0.40 = $500 = $156
The required return on Lone Stars levered equity is: rS = r0 + (B/S)(r0 rB)(1 TC) = 0.20 + ($500/$156)(0.20 0.10)(1 0.40) = 0.3923 Therefore, the required return on Lone Stars levered equity (rS) is 39.23%. d. In order to value a firms equity using the Flow-to-Equity approach, discount the cash flows available to equity holders at the cost of the firms levered equity (rS).
Answers to End-of-Chapter Problems
B-41
Since the pre-tax cost of debt is 10% and the firm has $500 of debt outstanding, Lone Star must pay $50 (= 0.10 * $500) in interest at the end of each year.
Lone Star Industries EBIT Interest Pre-Tax Earnings Taxes at 40% After-Tax Earnings $152.00 $50.00 $102.00 $40.80 $61.20
S = Cash Flows Available to Equity Holders / rS = After-Tax Earnings / rS = $61.20 / 0.3923 = $156 The value of Lone Stars equity is $156. 18.13 Use the Capital Asset Pricing Model to find the average cost of levered equity (rS) in the holiday gift industry. According to the Capital Asset Pricing Model: rS = rf + Equity(rm rf) where rS rf rm Equity = the cost of a firms levered equity = the risk-free rate = the expected return on the market portfolio = the firms equity beta
In this problem: rf = 0.09 rm = 0.17 Equity = 1.5 rS = rf + Equity(rm rf) = 0.09 + 1.5 (0.17-0.09) = 0.21 The average cost of levered equity in the holiday gift industry is 21%. Next, find the average cost of unlevered equity (r0) in the holiday gift industry. According to Modigliani-Miller Proposition II with corporate taxes rS = r0 + (B/S)(r0 rB)(1 TC) where r0 rS rB TC = the cost of unlevered equity = the cost of levered equity = the pre-tax cost of debt = the corporate tax rate
B-42
Answers to End-of-Chapter Problems
B/S = the firms target debt-to-equity ratio In this problem: rS = 0.21 rB = 0.10 TC = 0.40 B/S = 0.30 The average cost of unlevered equity in the holiday gift industry is: rS = r0 + (B/S)(r0 rB)(1 TC) 0.21= r0 + (.30)(r0 0.10)(1 0.40) r0 = 0.1932
The average cost of unlevered equity in the holiday gift industry is 19.32%. Next, use the average cost of unlevered equity in the holiday gift industry to find the cost of Blue Angels levered equity. According to Modigliani-Miller Proposition II with corporate taxes rS = r0 + (B/S)(r0 rB)(1 TC) where r0 = the cost of unlevered equity rS = the cost of levered equity rB = the pre-tax cost of debt TC = the corporate tax rate B/S = the firms target debt-to-equity ratio
In this problem: r0 = 0.1932 rB = 0.10 TC = 0.40 B/S = 0.35 The cost of Blue Angels levered equity is: rS = r0 + (B/S)(r0 rB)(1 TC) = 0.1932 + (0.35)(0.1932 0.10)(1 0.40) = 0.2128 The cost of Blue Angels levered equity is 21.28%. Since the project is financed at the firms target debt-equity ratio, it must be discounted at the Blue Angels weighted average cost of capital. In a world with corporate taxes, a firms weighted average cost of capital (rwacc) equals:
Answers to End-of-Chapter Problems B-43
rwacc where
= {B / (B+S)}(1 TC) rB + {S / (B+S)}rS B / (B+S) S / (B+S) rB rS TC = the firms debt-to-value ratio = the firms equity-to-value ratio = the pre-tax cost of debt = the cost of levered equity = the corporate tax rate
The problem does not provide either Blue Angels debt-to-value ratio or Blue Angels equity-to-value ratio. However, the firms debt-to-equity ratio of 0.35 is given, which can be written algebraically as: B / S = 0.35 Solving for B: B = (0.35)(S) A firms debt-to-value ratio is: Since B = (0.35)(S): Blue Angels debt-to-value ratio = (0.35)(S) / { (0.35)(S) + S} = (0.35)(S) / (1.35)(S) = (0.35)/(1.35) = 0.2593 Blue Angels debt-to-value ratio is 0.2593. A firms equity-to-value ratio is: S / (B+S) Since B = (0.35)(S): Blue Angels equity-to-value ratio = S / {(0.35)(S) + S} = S / (1.35S) = (1 / 1.35) = 0.7407 B / (B+S)
Blue Angels equity-to-value ratio is 0.7407. The inputs to the WACC calculation are: B / (B+S) S / (B+S) rB rS TC rwacc = 0.2593 = 0.7407 = 0.10 = 0.2128 = 0.40 = {B / (B+S)}(1 TC) rB + {S / (B+S)}rS = (0.2593)(1 0.40)(0.10) + (0.7407)(0.2128)
B-44
Answers to End-of-Chapter Problems
= 0.1732 Blue Angels weighted average cost of capital is 17.32%. Use the weighted average cost of capital to discount the projects cash flows. NPVPROJECT = -$325,000 + $55,000*GA5.1732, .05 + [$55,000(1.05)5 / .1732] / (1.1732)5 = $47,424
Since the NPV of the project is positive, Blue Angel should undertake the project. 18.14 a. If flotation costs are not taken into account, the net present value of a loan equals: NPVLoan = Gross Proceeds After-tax present value of interest and principal payments Proceeds net of flotation costs The gross proceeds of the loan are $4,250,000. After-tax present value of interest and principal payments Interest is paid off the gross proceeds of $4,250,000. Since the loan carries an interest rate of 9%, Kendrick will make interest payments of $382,500 [= (0.09)($4,250,000)] at the end of each year. However, since these payments are tax deductible, the after-tax cost of these payments is only $229,500 {= (1-.40)($382,500)} per year. At the end of ten years, Kendrick must repay the $4,250,000 in gross proceeds. Since the level of debt is known, the appropriate discount rate to use is Kendricks pre-tax cost of debt, 9%. After-Tax PV of Payments= $229,500A100.09 + $4,250,000 / (1.09)10 = $3,268,098 The after-tax present value of interest and principal payments is $3,268,098. NPVLoan = Gross Proceeds After-tax present value of interest and principal payments = $4,250,000 - $3,268,098 = $981,902 The NPV of the loan excluding flotation costs is $981,902. b. If flotation costs are taken into account, the net present value of a loan equals: NPVLoan = (Proceeds net of flotation costs) (After-tax present value of interest and principal payments) + (Present value of the flotation cost tax shield) Proceeds net of flotation costs The gross proceeds of the loan are $4,250,000. Flotation costs will be $53,125 (= 0.0125 * $4,250,000).
Answers to End-of-Chapter Problems B-45
Proceeds net of flotation costs are $4,196,875 (= $4,250,000 - $53,125). After-tax present value of interest and principal payments Interest is paid off the gross proceeds of $4,250,000. Since the loan carries an interest rate of 9%, Kendrick will make interest payments of $382,500 [= (0.09)($4,250,000)] at the end of each year. However, since these payments are tax deductible, the after-tax cost of these payments is only $229,500 {= (1-.40)($382,500)} per year. At the end of ten years, Kendrick must repay the $4,250,000 in gross proceeds. Since the level of debt is known, the appropriate discount rate to use is Kendricks pre-tax cost of debt, 9%. After-Tax PV of Payments= $229,500A100.09 + $4,250,000 / (1.09)10 = $3,268,098 The after-tax present value of interest and principal payments is $3,268,098. Present Value of the flotation cost tax shield Flotation costs will be amortized over the 10-year life of the loan, generating tax shields for Kendrick. Total flotation costs are $53,125 [= (0.0125)($4,250,000)]. Straight-line amortization of these costs over ten years yields annual flotation costs of $5,312.50 (= $53,125/10). The annual tax shield relating to these costs is: Annual Tax Shield = (TC)(Annual Flotation Expense) = (0.40)($5,312.50) = $2,125 PV(Flotation Cost Tax Shield) = $2,125A100.09 = $13,638 The present value of the flotation cost tax shield is $13,638. NPVLoan = (Proceeds net of flotation costs) (After-tax present value of interest and principal payments) + (Present value of the flotation cost tax shield) = $4,196,875 - $3,268,098 + $13,638 = $942,415 The NPV of the loan including flotation costs is $942,415. 18.15 a. The equity beta of a firm financed entirely by equity is equal to its unlevered beta. Find each of the firms equity betas, given an unlevered beta of 1.2. North Pole
Answers to End-of-Chapter Problems B-46
Equity = [1 + (1-TC)(B/S)]Unlevered where Equity Unlevered TC B S = the equity beta = the unlevered beta = the corporate tax rate = the value of the firms debt = the value of the firms equity
In this problem: Unlevered TC B S = 1.2 = 0.35 = $1,000,000 = $1,500,000
Equity = [1 + (1-TC)(B/S)]Unlevered = [1 + (1-0.35)($1,000,000/$1,500,000)][1.2] = 1.72 North Poles equity beta is 1.72. South Pole Equity = [1 + (1-TC)(B/S)]Unlevered where Equity Unlevered TC B S = the equity beta = the unlevered beta = the corporate tax rate = the value of the firms debt = the value of the firms equity
In this problem: Unlevered TC B S = 1.2 = 0.35 = $1,500,000 = $1,000,000
Equity = [1 + (1-TC)(B/S)]Unlevered = [1 + (1-0.35)($1,500,000/$1,000,000)][1.2] = 2.37 South Poles equity beta is 2.37. b. According to the Capital Asset Pricing Model: rS = rf + Equity(rm rf) where rS rf = the required rate of return on a firms equity = the risk-free rate
B-47
Answers to End-of-Chapter Problems
rm = the expected return on the market portfolio Equity = the equity beta North Pole: rf = 0.0425 rm = 0.1275 Equity = 1.72 rS = rf + Equity(rm rf) = 0.0425 + 1.72(0.1275-0.0425) = 0.1887 The required return on North Poles equity is 18.87%. South Pole: rf = 0.0425 rm = 0.1275 Equity = 2.37 rS = rf + Equity(rm rf) = 0.0425 + 2.37(0.1275-0.0425) = 0.2440 The required return on South Poles equity is 24.40%.
18.16
We have provided this solution for illustrative purposes only, as the data will change depending on the date that the data is accessed. Using the Financial Post Advisor , the following information was obtained: Effective income tax rate 46.2% (We have assumed that the effective rate is the same as the marginal rate.) The levered beta for MDS was found to be 0.78. The long term debt at October, 2004 was $488 million and shareholders equity was $1,497 million. For simplicity, we have assumed that the beta of debt equals zero in the formula. ( in reality, the beta of debt is likely greater than one, since the company is likely paying more than the risk free rate on its debt.)
D ( U D )(1 Tc ) E 488 0.78 U ( U 0)(1 .462) 1, 497 U 0.66
E U
Answers to End-of-Chapter Problems
B-48
18.17
We have provided this solution for illustrative purposes only, as the data will change depending on the date that the data is accessed. From the FP Advisor, we obtained the following information for BCE for the year end December, 2004. Book value of long term debt ( including current portion) Book value of Preferred shares Book value of common shares Total book value of firm Beta Effective tax rate ( assumes equal to the marginal tax rate) Common shares outstanding Share price Market capitalization Cost of debt ( using weighted average of interest rates from debt note provided) Cost of preferred shares Dividends / value = 70/1670 = In millions $ 12,930 1,670 12,362 26,962 0.84 29.81% 925,935,682 $30.24 28,000 7.44% 4.19%
Using the data from Table 10.2, we have calculated the cost of equity as follows:
Re Rf ( Rm Rf ) Re 6.80% 0.84(3.84%) 10.03%
The weighted average cost of capital using the book values is:
Debt PS ComEquity ( Rd )(1 Tc) ( RPS ) (Re) Value Value Value 12,930 1, 670 12,362 WACC (7.44%)(1 0.2981) (4.19%) (10.03%) 26,962 26,962 26,962 WACC 7.36% WACC
Assuming that the market value of debt and preferred shares is equal to their book values, and using the market capitalization value of the common shares, we have determined the firm value to be : 12,930 + 1,670 + 28,000 = 42,600. The revised WACC using market values is as follows:
Debt PS ComEquity ( Rd )(1 Tc) ( RPS ) (Re) Value Value Value 12,930 1, 670 28, 000 WACC (7.44%)(1 0.2981) (4.19%) (10.03%) 42, 600 42, 600 42, 600 WACC 8.34% WACC
Answers to End-of-Chapter Problems
B-49
The difference is that the current value of the common shares is significantly higher than its book value, resulting in a higher weighting of equity. Because the after tax cost of equity is higher than the debt, this results in a higher WACC using the market values. The correct method should be to use the current market values of the debt and equity. 18.18 We have provided this solution for illustrative purposes only, as the data will change depending on the date that the data is accessed. From the FP Advisor, we obtained the following current information for Manitoba telecom and Call-Net. Manitoba Tel In millions $ 916.1 1384.6 2300.7 0.75 28.43% 67,570,117 46.75 $3,158.9 4,075 5.76% Call-net In millions $ 268.5 177.4 445.9 1.28 28.43% 35,764,036 $6.90 246.8 515.3 10.625%
Book value of long term debt ( including current portion) Book value of common shares Total book value of firm Beta Effective tax rate ( assumes equal to the marginal tax rate) Common shares outstanding Share price Market capitalization Firm market value ( debt + market cap) Cost of debt ( using weighted average of interest rates from debt note provided)
For Call-Net, there was no effective tax rate since the company had losses. We have assumed a similar rate as Manitoba Telecom. Using the data from Table 10.2, we have calculated the cost of equity as follows:
Re Rf ( Rm Rf ) Manitoba Re 6.80% 0.75(3.84%) 9.68% Cal Re 6.80% 1.28(3.84%) 11.72%
The WACC for Manitoba Tel using market values is:
WACC
Debt ComEquity ( Rd )(1 Tc) (Re) Value Value
916.1 3,158.9 (5.76%)(1 0.2843) (9.68%) 4, 075 4, 075 WACC 8.43% WACC
Answers to End-of-Chapter Problems
B-50
The WACC of Call-net using market values was calculated as follows:
WACC
Debt ComEquity ( Rd )(1 Tc) (Re) Value Value
268.5 246.8 (10.625%)(1 0.2843) (11.72%) 515.3 515.3 WACC 9.58% WACC
Using the above information, we find the following WACCs: BCE 8.34% Manitoba Telecom 8.43% Call-Net 9.58% I comparing these, we find that the debt leverage for Manitoba Tel is the smallest, and therefore, its cost of equity is the lowest. Manitoba Tel is a large company but it is mainly operating in Western Canada. BCE is the largest company, but has a higher debt weighting than Manitoba Tel. Therefore, although its cost of equity is higher, the weighting of the debt cost brings the WACC slightly lower than Manitobas . Due its size and national coverage, BCE can likely take on more debt than Manitoba Telecom at this time. Call-Net is the smallest, and most volatile of the three competitors and this is seen in the significantly larger cost of equity and cost of debt that the company must pay. It has also had losses in recent years, making the debt and equity both more risky in comparison to BCE and Manitoba Tel. As a result, its cost of capital is the highest.
Answers to End-of-Chapter Problems
B-51