Cost of Capital Basic Problems 1-19

Problem 1

Calculating Cost of Equity

Question: The Tribiani Co. just issued a dividend of $2.90 per share on its common stock. The company is expected to maintain a constant 4.5 percent growth rate in its dividends indefinitely. If the stock sells for $56 a share, what is the company's cost of equity?

Theoretical Foundation: This problem uses the Dividend Growth Model to find the cost of equity, $R_E$. The formula is $R_E = \frac{D_1}{P_0} + g$, where $D_1$ is the next dividend payment, $P_0$ is the current price, and $g$ is the constant growth rate. The next dividend, $D_1$, is calculated as $D_0 \times (1 + g)$.

Step-by-step Solution:

First, identify the given variables: $D_0 = \$2.90$, $g = 4.5\%$, $P_0 = \$56$.
Calculate the expected next dividend, $D_1$: $D_1 = \$2.90 \times (1 + 0.045) = \$3.0305$.
Now, apply the Dividend Growth Model formula to find the cost of equity, $R_E$:
$R_E = \frac{\$3.0305}{\$56} + 0.045 = 0.0541 + 0.045 = \textbf{0.0991}$ or 9.91%

Problem 2

Calculating Cost of Equity

Question: The Swanson Corporation's common stock has a beta of 1.07. If the risk-free rate is 3.4 percent and the expected return on the market is 11 percent, what is the company's cost of equity capital?

Theoretical Foundation: This problem uses the Security Market Line (SML) approach, or CAPM, to calculate the cost of equity. The formula is $R_E = R_f + \beta_E \times (R_M - R_f)$, where $R_f$ is the risk-free rate, $R_M$ is the expected market return, and $\beta_E$ is the stock's beta. The term $(R_M - R_f)$ is known as the market risk premium.

Step-by-step Solution:

First, identify the given variables: $\beta_E = 1.07$, $R_f = 3.4\%$, $R_M = 11\%$.
Calculate the market risk premium: $R_M - R_f = 11\% - 3.4\% = 7.6\%$.
Apply the SML formula to find the cost of equity, $R_E$:
$R_E = 3.4\% + 1.07 \times 7.6\% = 3.4\% + 8.132\% = \textbf{11.532\%}$ or 11.53%

Problem 3

Calculating Cost of Equity

Question: Stock in Jansen Industries has a beta of 1.05. The market risk premium is 7 percent, and T-bills are currently yielding 3.5 percent. The company's most recent dividend was $2.45 per share, and dividends are expected to grow at an annual rate of 4.1 percent indefinitely. If the stock sells for $44 per share, what is your best estimate of the company's cost of equity?

Theoretical Foundation: This problem provides information for both the Dividend Growth Model and the SML approach. The best estimate is to calculate the cost of equity using both methods and then take the average of the two results, assuming they are reasonably close.

Step-by-step Solution:

Method 1: SML Approach
Given: $\beta_E = 1.05$, $(R_M - R_f) = 7\%$, $R_f = 3.5\%$.
$R_E = 3.5\% + 1.05 \times 7\% = 3.5\% + 7.35\% = \textbf{10.85\%}$

Method 2: Dividend Growth Model
Given: $D_0 = \$2.45$, $g = 4.1\%$, $P_0 = \$44$.
$D_1 = \$2.45 \times (1 + 0.041) = \$2.5505$.
$R_E = \frac{\$2.5505}{\$44} + 0.041 = 0.05796 + 0.041 = \textbf{0.09896}$ or 9.90%

Since the two estimates are relatively close, we can average them to find the best estimate.

Best Estimate: $ \frac{10.85\% + 9.90\%}{2} = \frac{20.75\%}{2} = \textbf{10.375\%} $

Problem 4

Estimating the DCF Growth Rate

Question: Suppose Wacken, Ltd., just issued a dividend of $2.73 per share on its common stock. The company paid dividends of $2.31, $2.39, $2.48, and $2.58 per share in the last four years. If the stock currently sells for $43, what is your best estimate of the company's cost of equity capital using the arithmetic average growth rate in dividends? What if you use the geometric average growth rate?

Theoretical Foundation: The cost of equity using the Dividend Growth Model is highly sensitive to the estimated growth rate, $g$. We can estimate $g$ using historical data by calculating either the arithmetic or the geometric average growth rate. The arithmetic average is the simple average of annual percentage changes, while the geometric average represents the constant compounded growth rate over the period.

Step-by-step Solution:

First, calculate the annual dividend growth rates:
$g_1 = \frac{2.39 - 2.31}{2.31} = 3.46\%$
$g_2 = \frac{2.48 - 2.39}{2.39} = 3.77\%$
$g_3 = \frac{2.58 - 2.48}{2.48} = 4.03\%$
$g_4 = \frac{2.73 - 2.58}{2.58} = 5.81\%$

Arithmetic Average Growth Rate:
$g_{arithmetic} = \frac{3.46\% + 3.77\% + 4.03\% + 5.81\%}{4} = \textbf{4.27\%}$
Cost of Equity: $R_E = \frac{\$2.73 \times (1+0.0427)}{\$43} + 0.0427 = \frac{\$2.8465}{\$43} + 0.0427 = 0.0662 + 0.0427 = \textbf{0.1089}$ or 10.89%

Geometric Average Growth Rate:
Geometric growth rate is found by solving for $g$ in the equation: $D_{current} = D_{past} \times (1 + g)^n$.
$\$2.73 = \$2.31 \times (1 + g)^4$
$(1 + g)^4 = \frac{2.73}{2.31} = 1.1818$
$1 + g = 1.1818^{1/4} = 1.0425$
$g_{geometric} = 0.0425$ or 4.25%
Cost of Equity: $R_E = \frac{\$2.73 \times (1+0.0425)}{\$43} + 0.0425 = \frac{\$2.8450}{\$43} + 0.0425 = 0.0662 + 0.0425 = \textbf{0.1087}$ or 10.87%

Problem 5

Calculating Cost of Preferred Stock

Question: Savers has an issue of preferred stock with a stated dividend of $3.85 that just sold for $87 per share. What is the bank's cost of preferred stock?

Theoretical Foundation: Preferred stock is treated as a perpetuity, paying a fixed dividend forever. Its cost, $R_P$, is the dividend yield, calculated as the annual dividend divided by the current stock price.

Step-by-step Solution:

Identify the given variables: Dividend, $D = \$3.85$, Price, $P_0 = \$87$.
Apply the cost of preferred stock formula:
$R_P = \frac{D}{P_0} = \frac{\$3.85}{\$87} = \textbf{0.0443}$ or 4.43%

Problem 6

Calculating Cost of Debt

Question: Sunrise, Inc., is trying to determine its cost of debt. The firm has a debt issue outstanding with 23 years to maturity that is quoted at 96 percent of face value. The issue makes semiannual payments and has an embedded cost of 5 percent annually. What is the company's pretax cost of debt? If the tax rate is 21 percent, what is the aftertax cost of debt?

Theoretical Foundation: The pretax cost of debt, $R_D$, is the yield to maturity (YTM) on new debt. The YTM is the discount rate that equates the present value of a bond's future cash flows (coupons and face value) to its current market price. The aftertax cost of debt is calculated as $R_D \times (1 - T_C)$.

Step-by-step Solution:

First, find the YTM. The bond has a face value of $1,000, is quoted at 96% of face value, and has an annual coupon rate of 5%. The payments are semiannual.
Number of periods, $N = 23 \text{ years} \times 2 = 46$ periods.
Annual coupon payment = $0.05 \times \$1,000 = \$50$. Semiannual coupon payment, $C = \$25$.
Current price, $P_0 = 0.96 \times \$1,000 = \$960$.
Face Value, $FV = \$1,000$.
Using a financial calculator or spreadsheet, solve for the semiannual yield, $r$, in the bond pricing formula: $ \$960 = \sum_{t=1}^{46} \frac{\$25}{(1+r)^t} + \frac{\$1,000}{(1+r)^{46}} $
Solving for $r$, we get $r \approx 2.65\%$.
Pretax Cost of Debt, $R_D = 2.65\% \times 2 = \textbf{5.30\%}$.
Now, calculate the aftertax cost of debt: $R_D \times (1 - T_C)$.
Aftertax Cost of Debt: $5.30\% \times (1 - 0.21) = 5.30\% \times 0.79 = \textbf{4.197\%}$ or 4.20%

Problem 7

Calculating Cost of Debt

Question: Jiminy's Cricket Farm issued a 30-year, 4.5 percent semiannual bond three years ago. The bond currently sells for 104 percent of its face value. The company's tax rate is 22 percent.

a. What is the pretax cost of debt?

b. What is the aftertax cost of debt?

c. Which is more relevant, the pretax or the aftertax cost of debt? Why?

Theoretical Foundation: The pretax cost of debt is the current yield to maturity (YTM). The aftertax cost of debt is the pretax cost adjusted for the tax shield on interest payments. The aftertax cost of debt is the relevant rate for WACC calculations because we are concerned with aftertax cash flows and costs.

Step-by-step Solution:

The bond was issued 3 years ago with a 30-year maturity, so there are $30 - 3 = 27$ years remaining.
The bond makes semiannual payments. $N = 27 \times 2 = 54$ periods.
Annual coupon payment = $0.045 \times \$1,000 = \$45$. Semiannual coupon, $C = \$22.50$.
Current price, $P_0 = 1.04 \times \$1,000 = \$1,040$.
Face Value, $FV = \$1,000$.
a. Pretax Cost of Debt: Solve for the semiannual yield, $r$, in the bond pricing formula: $ \$1,040 = \sum_{t=1}^{54} \frac{\$22.50}{(1+r)^t} + \frac{\$1,000}{(1+r)^{54}} $
Solving for $r$, we get $r \approx 2.115\%$. Pretax cost, $R_D = 2.115\% \times 2 = \textbf{4.23\%}$.
b. Aftertax Cost of Debt: Aftertax cost = $4.23\% \times (1 - T_C)$
$ = 4.23\% \times (1 - 0.22) = 4.23\% \times 0.78 = \textbf{3.30\%}$
c. Relevance: The **aftertax cost of debt** is more relevant for capital budgeting and WACC calculations because it reflects the actual cost of debt to the firm after considering the tax-deductible nature of interest payments.

Problem 8

Calculating Cost of Debt

Question: For the firm in Problem 7, suppose the book value of the debt issue is $75 million. In addition, the company has a second debt issue on the market, a zero coupon bond with eight years left to maturity; the book value of this issue is $30 million, and the bonds sell for 81 percent of par. What is the company's total book value of debt? The total market value? What is your best estimate of the aftertax cost of debt now?

Theoretical Foundation: The total market value of debt is the sum of the market values of all outstanding debt issues. The best estimate of the aftertax cost of debt is the weighted average of the aftertax yields of all debt issues, using their market values as weights.

Step-by-step Solution:

Total Book Value of Debt:
Book Value = $ \$75 \text{ million} + \$30 \text{ million} = \textbf{\$105 million} $

Total Market Value of Debt:
From Problem 7, the first bond issue sells at 104% of par. To find the market value, we first need to find its face value. Assuming its book value is equal to its face value, then its market value is $ \$75 \text{ million} \times 1.04 = \$78 \text{ million} $.
The second bond issue is a zero-coupon bond selling for 81% of par. Its book value is given as $30M, so its market value is $ \$30 \text{ million} \times 0.81 = \$24.3 \text{ million} $.
Total Market Value = $ \$78 \text{ million} + \$24.3 \text{ million} = \textbf{\$102.3 million} $

Aftertax Cost of Debt:
We need to find the YTM for the zero-coupon bond. $ \$24.3 = \frac{\$30}{(1+r)^8} \implies (1+r)^8 = \frac{30}{24.3} = 1.2346 $.
$ r = 1.2346^{1/8} - 1 = 0.0269$ or $2.69\%$. This is the annual YTM.
Aftertax YTM for zero-coupon bond = $2.69\% \times (1 - 0.22) = 2.10\%$.
Aftertax YTM for first bond = $3.30\%$ (from Problem 7).
Weighted Average Aftertax Cost = $ \frac{\$78}{\$102.3} \times 3.30\% + \frac{\$24.3}{\$102.3} \times 2.10\% = 2.51\% + 0.50\% = \textbf{3.01\%} $

Problem 9

Calculating WACC

Question: Ninecent Corporation has a target capital structure of 70 percent common stock, 5 percent preferred stock, and 25 percent debt. Its cost of equity is 11 percent, the cost of preferred stock is 5 percent, and the pretax cost of debt is 6 percent. The relevant tax rate is 23 percent.

a. What is the company's WACC?

b. The company president has approached you about the company's capital structure. He wants to know why the company doesn't use more preferred stock financing because it costs less than debt. What would you tell the president?

Theoretical Foundation: The WACC is the weighted average of the aftertax costs of all capital sources. The aftertax cost of debt is used because of the tax-deductible nature of interest payments. The cost of preferred stock, while seemingly low, is not tax-deductible and has other characteristics that make it less desirable than debt.

Step-by-step Solution:

a. WACC Calculation:
Given: $E/V = 0.70$, $P/V = 0.05$, $D/V = 0.25$.
$R_E = 11\%$, $R_P = 5\%$, $R_D = 6\%$. Tax rate, $T_C = 23\%$.
$WACC = (0.70 \times 0.11) + (0.05 \times 0.05) + (0.25 \times 0.06 \times (1 - 0.23))$
$ = 0.077 + 0.0025 + 0.01155 = \textbf{0.09105}$ or 9.11%

b. Explanation to the President:
The president's reasoning is flawed. While the pretax cost of preferred stock (5%) is lower than the pretax cost of debt (6%), the cost of debt is tax-deductible. The aftertax cost of debt is $6\% \times (1-0.23) = 4.62\%$.
Therefore, the aftertax cost of debt (4.62%) is actually **lower** than the cost of preferred stock (5%). Additionally, preferred stock dividends are not tax-deductible, which is a significant disadvantage compared to debt.

Problem 10

Taxes and WACC

Question: Brannan Manufacturing has a target debt-equity ratio of .35. Its cost of equity is 11 percent, and its pretax cost of debt is 6 percent. If the tax rate is 21 percent, what is the company's WACC?

Theoretical Foundation: The WACC formula requires the capital structure weights $E/V$ and $D/V$. Given the debt-equity ratio, $D/E$, we can convert it into these weights. For a given $D/E$ ratio, $E/V = \frac{1}{1 + D/E}$ and $D/V = \frac{D/E}{1 + D/E}$.

Step-by-step Solution:

Given: $D/E = 0.35$.
Calculate the capital structure weights:
$E/V = \frac{1}{1 + 0.35} = \frac{1}{1.35} \approx 0.7407$
$D/V = \frac{0.35}{1.35} \approx 0.2593$
Given: $R_E = 11\%$, $R_D = 6\%$, $T_C = 21\%$.
Apply the WACC formula: $WACC = (0.7407 \times 0.11) + (0.2593 \times 0.06 \times (1 - 0.21))$
$ = 0.081477 + 0.012297 = \textbf{0.0938}$ or 9.38%

Problem 11

Finding the Target Capital Structure

Question: Fama's Llamas has a weighted average cost of capital of 8.4 percent. The company's cost of equity is 11 percent, and its pretax cost of debt is 5.8 percent. The tax rate is 25 percent. What is the company's target debt-equity ratio?

Theoretical Foundation: We can work backward from the WACC formula to find the capital structure weights. Once we have the weights, we can easily find the debt-equity ratio using the formula $D/E = \frac{D/V}{E/V}$.

Step-by-step Solution:

Given: $WACC = 8.4\%$, $R_E = 11\%$, $R_D = 5.8\%$, $T_C = 25\%$.
Let $x_E$ be the weight of equity and $x_D$ be the weight of debt. We know $x_E + x_D = 1$, so $x_D = 1 - x_E$.
Set up the WACC equation: $0.084 = x_E(0.11) + (1 - x_E)(0.058)(1 - 0.25)$
$0.084 = 0.11x_E + (1 - x_E)(0.0435)$
$0.084 = 0.11x_E + 0.0435 - 0.0435x_E$
$0.0405 = 0.0665x_E$
$x_E = 0.0405 / 0.0665 \approx 0.609$
Since $x_E = E/V$, the weight of equity is 60.9%. The weight of debt is $x_D = 1 - 0.609 = 0.391$.
Target Debt-Equity Ratio: $D/E = \frac{x_D}{x_E} = \frac{0.391}{0.609} \approx \textbf{0.64}$

Problem 12

Book Value versus Market Value

Question: Dani Corp. has 5.5 million shares of common stock outstanding. The current share price is $83, and the book value per share is $5. The company also has two bond issues outstanding. The first bond issue has a face value of $80 million, a coupon rate of 5.5 percent, and sells for 109 percent of par. The second issue has a face value of $45 million, a coupon rate of 5.8 percent, and sells for 108 percent of par. The first issue matures in 21 years, the second in 6 years. Both bonds make semiannual coupon payments.

a. What are the company's capital structure weights on a book value basis?

b. What are the company's capital structure weights on a market value basis?

c. Which are more relevant, the book or market value weights? Why?

Theoretical Foundation: WACC calculations require market value weights, as they reflect the true, current value of the firm's financing sources. Book values can be misleading, especially for equity, as they often do not reflect the current market value.

Step-by-step Solution:

a. Book Value Weights:
Book Value of Equity = $5.5M \times \$5 = \$27.5M$.
Book Value of Debt = $ \$80M + \$45M = \$125M $.
Total Book Value = $ \$27.5M + \$125M = \$152.5M $.
Book Value Weight of Equity: $\frac{\$27.5M}{\$152.5M} = \textbf{0.1803}$ or 18.03%.
Book Value Weight of Debt: $\frac{\$125M}{\$152.5M} = \textbf{0.8197}$ or 81.97%.

b. Market Value Weights:
Market Value of Equity = $5.5M \times \$83 = \$456.5M$.
Market Value of Debt 1 = $ \$80M \times 1.09 = \$87.2M $.
Market Value of Debt 2 = $ \$45M \times 1.08 = \$48.6M $.
Total Market Value of Debt = $ \$87.2M + \$48.6M = \$135.8M $.
Total Market Value = $ \$456.5M + \$135.8M = \$592.3M $.
Market Value Weight of Equity: $\frac{\$456.5M}{\$592.3M} = \textbf{0.7707}$ or 77.07%.
Market Value Weight of Debt: $\frac{\$135.8M}{\$592.3M} = \textbf{0.2293}$ or 22.93%.

c. Which are more relevant?
Market value weights are more relevant. They reflect the current value of the firm's capital sources, which is what investors are actually contributing to the firm today. The book value of equity is particularly misleading as it is based on historical accounting values and does not reflect current market conditions.

Problem 13

Calculating the WACC

Question: In Problem 12, suppose the most recent dividend was $3.85 and the dividend growth rate is 5 percent. Assume that the overall cost of debt is the weighted average of that implied by the two outstanding debt issues. The tax rate is 21 percent. What is the company's WACC?

Theoretical Foundation: To find the WACC, we need the costs of equity and debt, and the market value weights. The cost of equity can be found using the Dividend Growth Model. The cost of debt is the weighted average of the YTMs of the two bond issues. Finally, the WACC is the weighted average of these costs, adjusted for taxes.

Step-by-step Solution:

Cost of Equity ($R_E$):
$D_0 = \$3.85$, $g = 5\%$, $P_0 = \$83$.
$D_1 = \$3.85 \times (1.05) = \$4.0425$.
$R_E = \frac{\$4.0425}{\$83} + 0.05 = 0.0487 + 0.05 = \textbf{0.0987}$ or 9.87%.

Cost of Debt ($R_D$):
First, find the YTM for each bond. The aftertax cost of debt will be the weighted average of these YTMs.
Bond 1: $N = 42$, $C = \$27.50$, $FV = \$1,000$, $P_0 = \$1,090$. Semiannual YTM $r_1 \approx 2.37\%$. Annual $R_{D1} = 4.74\%$.
Bond 2: $N = 12$, $C = \$29$, $FV = \$1,000$, $P_0 = \$1,080$. Semiannual YTM $r_2 \approx 4.09\%$. Annual $R_{D2} = 8.18\%$.
Weighted Average Cost of Debt: $R_D = \frac{\$87.2M}{\$135.8M} \times 4.74\% + \frac{\$48.6M}{\$135.8M} \times 8.18\% = 0.642 \times 4.74\% + 0.358 \times 8.18\% = 3.04\% + 2.93\% = \textbf{5.97\%}$.

WACC:
From Problem 12, $E/V = 0.7707$ and $D/V = 0.2293$.
$WACC = (0.7707 \times 9.87\%) + (0.2293 \times 5.97\% \times (1 - 0.21))$
$ = 7.605\% + 1.08\% = \textbf{8.685\%}$ or 8.69%.

Problem 14

WACC

Question: Ursala, Inc., has a target debt-equity ratio of .65. Its WACC is 10.4 percent, and the tax rate is 23 percent.

a. If the company's cost of equity is 14 percent, what is its pretax cost of debt?

b. If instead you know that the aftertax cost of debt is 5.8 percent, what is the cost of equity?

Theoretical Foundation: The WACC formula can be rearranged to solve for any of the unknown variables if the others are known. We will first convert the debt-equity ratio to the capital structure weights.

Step-by-step Solution:

Given: $D/E = 0.65$.
$E/V = \frac{1}{1 + 0.65} = 0.6061$. $D/V = \frac{0.65}{1.65} = 0.3939$.

a. Find $R_D$:
Given: $WACC = 10.4\%$, $R_E = 14\%$, $T_C = 23\%$.
$10.4\% = (0.6061 \times 14\%) + (0.3939 \times R_D \times (1 - 0.23))$
$10.4\% = 8.4854\% + 0.3033R_D$
$1.9146\% = 0.3033R_D$
$R_D = \frac{0.019146}{0.3033} = \textbf{0.0631}$ or 6.31%.

b. Find $R_E$:
Given aftertax cost of debt: $R_D(1 - T_C) = 5.8\%$.
$10.4\% = (0.6061 \times R_E) + (0.3939 \times 5.8\%)$
$10.4\% = 0.6061R_E + 2.2846\%$
$8.1154\% = 0.6061R_E$
$R_E = \frac{0.081154}{0.6061} = \textbf{0.1339}$ or 13.39%.

Problem 15

Finding the WACC

Question: Given the following information for Lightning Power Co., find the WACC. Assume the company's tax rate is 21 percent.

Debt: 12,000 bonds with a 4.6 percent coupon outstanding, $1,000 par value, 25 years to maturity, selling for 105 percent of par; the bonds make semiannual payments.

Common stock: 575,000 shares outstanding, selling for $81 per share; the beta is 1.04.

Preferred stock: 30,000 shares of 3.4 percent preferred stock outstanding, a $100 par value, currently selling for $94 per share.

Market: 7 percent market risk premium and 3.2 percent risk-free rate.

Theoretical Foundation: This is a comprehensive WACC calculation that requires finding the cost and market value for each of the three capital components: common equity, preferred stock, and debt. Then, use the market value weights and aftertax costs to calculate the WACC.

Step-by-step Solution:

1. Market Values:
Market Value of Debt, $D$: $12,000 \times (\$1,000 \times 1.05) = \textbf{\$12.6M}$.
Market Value of Equity, $E$: $575,000 \times \$81 = \textbf{\$46.575M}$.
Market Value of Preferred Stock, $P$: $30,000 \times \$94 = \textbf{\$2.82M}$.
Total Market Value, $V$: $12.6M + 46.575M + 2.82M = \textbf{\$61.995M}$.

2. Weights:
$D/V = \frac{12.6M}{61.995M} = \textbf{0.2032}$.
$E/V = \frac{46.575M}{61.995M} = \textbf{0.7513}$.
$P/V = \frac{2.82M}{61.995M} = \textbf{0.0455}$.

3. Costs:
Cost of Debt ($R_D$): Find YTM for a bond with $N=50$, $P_0=\$1,050$, $C=\$23$. Semiannual $r \approx 2.147\%$, so annual $R_D = 4.294\%$.
Cost of Equity ($R_E$): Using SML, $R_E = 3.2\% + 1.04 \times 7\% = 3.2\% + 7.28\% = \textbf{10.48\%}$.
Cost of Preferred Stock ($R_P$): $R_P = \frac{D}{P_0} = \frac{0.034 \times \$100}{\$94} = \frac{\$3.40}{\$94} = \textbf{3.62\%}$.

4. WACC Calculation:
$WACC = (0.7513 \times 10.48\%) + (0.0455 \times 3.62\%) + (0.2032 \times 4.294\% \times (1 - 0.21))$
$ = 0.0787 + 0.00165 + 0.00683 = \textbf{0.0872}$ or 8.72%.

Problem 16

Finding the WACC

Question: Lingenburger Cheese Corporation has 6.4 million shares of common stock outstanding, 200,000 shares of 3.8 percent preferred stock outstanding, par value of $100, and 120,000 bonds with a semiannual coupon of 4.8 percent outstanding, par value $1,000 each. The common stock currently sells for $54 per share and has a beta of 1.08, the preferred stock currently sells for $103 per share, and the bonds have 15 years to maturity and sell for 107 percent of par. The market risk premium is 7.5 percent, T-bills are yielding 2.4 percent, and the company's tax rate is 22 percent.

a. What is the firm's market value capital structure?

b. If the company is evaluating a new investment project that has the same risk as the firm's typical project, what rate should the firm use to discount the project's cash flows?

Theoretical Foundation: This problem is a comprehensive WACC calculation. First, find the market value of each component to determine the capital structure weights. Then, calculate the cost of each component, and finally, compute the WACC using the market value weights and aftertax costs. The WACC is the appropriate discount rate for projects with the same risk as the firm's existing operations.

Step-by-step Solution:

1. Market Values:
Market Value of Equity, $E$: $6.4M \times \$54 = \textbf{\$345.6M}$.
Market Value of Preferred Stock, $P$: $200,000 \times \$103 = \textbf{\$20.6M}$.
Market Value of Debt, $D$: $120,000 \times (\$1,000 \times 1.07) = \textbf{\$128.4M}$.
Total Market Value, $V$: $345.6M + 20.6M + 128.4M = \textbf{\$494.6M}$.

a. Market Value Capital Structure (Weights):
$E/V = \frac{345.6M}{494.6M} = \textbf{0.6987}$.
$P/V = \frac{20.6M}{494.6M} = \textbf{0.0416}$.
$D/V = \frac{128.4M}{494.6M} = \textbf{0.2596}$.

2. Costs:
Cost of Debt ($R_D$): Find YTM for a bond with $N=30$, $P_0=\$1,070$, $C=\$24$. Semiannual $r \approx 2.11\%$, so annual $R_D = 4.22\%$.
Cost of Equity ($R_E$): Using SML, $R_E = 2.4\% + 1.08 \times 7.5\% = 2.4\% + 8.1\% = \textbf{10.5\%}$.
Cost of Preferred Stock ($R_P$): $R_P = \frac{D}{P_0} = \frac{0.038 \times \$100}{\$103} = \frac{\$3.80}{\$103} = \textbf{3.69\%}$.

b. Discount Rate (WACC):
$WACC = (0.6987 \times 10.5\%) + (0.0416 \times 3.69\%) + (0.2596 \times 4.22\% \times (1 - 0.22))$
$ = 0.07336 + 0.00153 + 0.00854 = \textbf{0.0834}$ or 8.34%.

The firm should use a discount rate of **8.34%** for a new investment project that has the same risk as its typical project.

Problem 17

SML and WACC

Question: An all-equity firm is considering the following projects:

Project	Beta	IRR
W	.83	9.4%
X	.92	11.6%
Y	1.09	12.9%
Z	1.35	14.1%

The T-bill rate is 4 percent, and the expected return on the market is 12 percent.

a. Which projects have a higher expected return than the firm's 12 percent cost of capital?

b. Which projects should be accepted?

c. Which projects would be incorrectly accepted or rejected if the firm's overall cost of capital were used as a hurdle rate?

Theoretical Foundation: The appropriate hurdle rate for a project is its required return, as determined by the SML (CAPM), not the firm's overall WACC unless the project's risk (beta) is the same as the firm's. A project should be accepted if its IRR is greater than its required return (i.e., it plots above the SML).

Step-by-step Solution:

The firm is all-equity, so WACC = $R_E$. The firm's beta is assumed to be the market beta, 1.0.
Firm's Cost of Capital ($R_E$): $R_E = 4\% + 1.0 \times (12\% - 4\%) = 12\%$.
The required return for each project is determined by the SML: $R_{Project} = 4\% + \beta_{Project} \times 8\%$.
Required return for W: $4\% + 0.83 \times 8\% = 10.64\%$.
Required return for X: $4\% + 0.92 \times 8\% = 11.36\%$.
Required return for Y: $4\% + 1.09 \times 8\% = 12.72\%$.
Required return for Z: $4\% + 1.35 \times 8\% = 14.80\%$.

a. Projects with IRR > Firm's WACC:
WACC = 12%. Projects with IRR > 12% are **Y (12.9%)** and **Z (14.1%)**.

b. Projects that should be accepted (IRR > Required Return):
Project W: $9.4\% < 10.64\%$ (Reject)
Project X: $11.6\% > 11.36\%$ (Accept)
Project Y: $12.9\% > 12.72\%$ (Accept)
Project Z: $14.1\% < 14.80\%$ (Reject)
Projects to be accepted: **X and Y**.

c. Incorrect decisions with WACC hurdle rate:
Project W (low risk): IRR of 9.4% is less than WACC of 12%. Correctly rejected.
Project X (low risk): IRR of 11.6% is less than WACC of 12%. **Incorrectly rejected**. Its true required return is only 11.36%.
Project Y (high risk): IRR of 12.9% is greater than WACC of 12%. **Incorrectly accepted**. Its true required return is 12.72%, so this decision is correct. Wait, the IRR is 12.9% and the required return is 12.72%. The difference is positive. So the decision is correct. Let me double check that. Project Y: 12.9% IRR is greater than 12.72% required return. Accept. Project Y: 12.9% IRR is greater than WACC of 12%. Accept. So the decision is correct.
Project Z (high risk): IRR of 14.1% is greater than WACC of 12%. **Incorrectly accepted**. Its true required return is 14.80%.
Wait, let me double check Project Y. Project Y: 12.9% IRR. WACC 12%. Required return 12.72%. IRR > Required return. Correct decision: accept. IRR > WACC. Accept. So no incorrect decision is made for Project Y.
The problem here is subtle. The question asks which projects would be incorrectly accepted or rejected. Project Y has an IRR of 12.9%, which is greater than the firm's WACC of 12%, so it would be accepted. Its true required return is 12.72%, so the acceptance is correct. So no incorrect decision is made. Wait, my previous answer for Y was incorrect. Let me re-examine.
Let's re-examine my previous response logic. I said `Projects to be accepted: **X and Y**`. This is correct. The next part `Projects with IRR > Firm's WACC: Y and Z`. This is also correct. `Projects that should be accepted (IRR > Required Return): X and Y`. This is correct. `Incorrect decisions with WACC hurdle rate:`. Project X has IRR 11.6% which is < WACC 12%. So it would be rejected. But its required return is 11.36%. IRR > required return. So it should be accepted. So `X is incorrectly rejected`. Project Z has IRR 14.1% which is > WACC 12%. So it would be accepted. But its required return is 14.8%. IRR < required return. So it should be rejected. So `Z is incorrectly accepted`. Project Y has IRR 12.9% and WACC 12%. It is accepted. The required return is 12.72%. IRR > required return. So it is correctly accepted. Project W has IRR 9.4% and WACC 12%. It is rejected. The required return is 10.64%. IRR < required return. So it is correctly rejected. So, the only incorrect decisions are for projects **X** and **Z**.
**X would be incorrectly rejected.**
**Z would be incorrectly accepted.**

Problem 18

Calculating Flotation Costs

Question: Suppose your company needs $43 million to build a new assembly line. Your target debt-equity ratio is .75. The flotation cost for new equity is 6 percent, but the flotation cost for debt is only 2 percent. Your boss has decided to fund the project by borrowing money because the flotation costs are lower and the needed funds are relatively small.

a. What do you think about the rationale behind borrowing the entire amount?

b. What is your company's weighted average flotation cost, assuming all equity is raised externally?

c. What is the true cost of building the new assembly line after taking flotation costs into account? Does it matter in this case that the entire amount is being raised from debt?

Theoretical Foundation: The flotation cost should be calculated based on the company's target capital structure, not the financing for a single project. The true cost of a project is its initial cost divided by $ (1 - f_A) $, where $ f_A $ is the weighted average flotation cost. It matters that the entire amount is raised from debt because this decision goes against the target capital structure and would need to be rebalanced later, making the weighted average flotation cost the correct approach.

Step-by-step Solution:

a. Rationale for borrowing the entire amount:
The rationale is incorrect. A company should make its financing decisions to maintain its target capital structure over the long run. Financing a single project with all debt will increase the debt-equity ratio, and the company will have to raise equity later to rebalance, incurring equity flotation costs anyway. The correct approach is to use the weighted average flotation cost based on the target capital structure.

b. Weighted average flotation cost:
Given $D/E = 0.75$. This means $E/V = \frac{1}{1 + 0.75} = 0.5714$ and $D/V = \frac{0.75}{1.75} = 0.4286$.
$f_A = (0.5714 \times 0.06) + (0.4286 \times 0.02) = 0.034284 + 0.008572 = \textbf{0.042856}$ or 4.29%.

c. True cost of the new assembly line:
True Cost = $ \frac{\text{Project Cost}}{1 - f_A} = \frac{\$43M}{1 - 0.0429} = \frac{\$43M}{0.9571} = \textbf{\$44.92M} $.
It matters that the entire amount is raised from debt because using the project-specific financing decision would incorrectly exclude the higher cost of equity flotation, leading to a flawed analysis. The weighted average approach correctly captures the long-term costs of maintaining the target capital structure.

Problem 19

Calculating Flotation Costs

Question: Shinedown Company needs to raise $95 million to start a new project and will raise the money by selling new bonds. The company will generate no internal equity for the foreseeable future. The company has a target capital structure of 65 percent common stock, 5 percent preferred stock, and 30 percent debt. Flotation costs for issuing new common stock are 7 percent, for new preferred stock, 4 percent, and for new debt, 2 percent. What is the true initial cost figure the company should use when evaluating its project?

Theoretical Foundation: The true initial cost of a project is the project cost plus the flotation costs. The flotation cost must be calculated as a weighted average based on the company's target capital structure, even if a single source of financing is used for the project. The formula for the weighted average flotation cost must include all capital components.

Step-by-step Solution:

First, calculate the weighted average flotation cost, $f_A$.
$f_A = (E/V \times f_E) + (P/V \times f_P) + (D/V \times f_D)$
$ = (0.65 \times 0.07) + (0.05 \times 0.04) + (0.30 \times 0.02) $
$ = 0.0455 + 0.002 + 0.006 = \textbf{0.0535}$ or 5.35%.
Now, calculate the true initial cost figure for the project, accounting for flotation costs.
True Initial Cost = $ \frac{\text{Project Amount}}{1 - f_A} = \frac{\$95M}{1 - 0.0535} = \frac{\$95M}{0.9465} = \textbf{\$100.37M} $.