Chapter 12, Problem 25QP

a)

Summary Introduction

To determine: The initial cash flow of the project at Time 0

Introduction:

Project evaluation refers to the process of evaluating the project for its feasibility and economic viability.

Answer to Problem 25QP

The initial cash flow at Time 0 is $20,825,000.

Explanation of Solution

Given information:

Company D manufactures radar detection systems. It is planning to open a new manufacturing plant overseas. The project will take five years. It had purchased a land overseas at $4,500,000 for dumping chemical waste. However, the company found an alternative to dispose of waste and the land remained unused.

The current after-tax valuation of the land is $5,000,000. It would be worth $5,300,000 in five years. The company wants to implement the project on this unused land. The cost of plant and equipment would be $15,000,000. The current market data for Company D is as follows:

• Debt: Company D has bonds with a face value of $1,000 and a coupon rate of 6.2 percent. There are 40,000 outstanding bonds. The bonds will mature in 25 years, and they make semiannual coupon payments. The bond currently sells at 95 percent of the face value.
• Common equity: The common equity of Company D has 825,000 shares outstanding. The current selling price of the share is $97. The stock has a beta of 1.15. The risk-free rate is 3.8 percent, and the market risk premium is 7 percent.
• Preferred stock: Company D has 45,000 outstanding preferred shares. The dividend is 5.8 percent, and the current market value is $95 per share. Assume that the face value of the share is $100.

The tax rate applicable to Company D is 34 percent. The project needs an initial net working capital of $825,000 to begin the operations.

Explanation:

Compute the initial cash outflow:

The initial cash flow includes the cost of the land, plant, and equipment, and the initial net working capital. The land currently values at $5,000,000. The cost of buying plant and equipment is $15,000,000. The initial working capital required is $825,000. Hence, the initial cash outflow at Time 0 is $20,825,000 $\left(\$5,000,000+\$15,000,000+\$825,000\right)$

b)

Summary Introduction

To determine: The discount rate

Introduction:

Project evaluation refers to the process of evaluating the project for its feasibility and economic viability.

Answer to Problem 25QP

The discount rate is 11.31 percent.

Explanation of Solution

The company can use the weighted average cost of capital as its discount rate if the new project’s risk is similar to the company’s risk. It is given that the project is riskier than the firm is. The company adds 2 percent to its weighted average cost of capital to match the riskiness of the project. Therefore, the discount rate for the project would we equal to the sum of the weighted average cost of capital and the adjustment factor of 2 percent.

The formula to calculate the market value of debt:

$\text{Market value of debt}=\text{Face value of the debt}\times \text{Price quote}$

The formula to calculate the market value of equity:

$\text{Market value of equity }(E)=\text{Outstanding equity shares}\times \text{Market value per share}$

The formula to calculate the market value of preferred stock:

$\begin{array}{l}\text{Market value of}\\ \text{preferred shares }(P)\end{array}\}=\text{Outstanding preferred shares}\times \text{Market value per share}$

The formula to calculate the total market value of the capital structure:

$\text{Total market value }(V)=\left(\begin{array}{l}\text{Market value}\\ \text{of equity }(E)\end{array}\right)+\left(\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of debt }(D)\end{array}\right)$

The formula to calculate the cost of equity using the Security market line (SML) approach:

$R_E=R_f+\left[R_M-R_f\right]\times {\beta }_E$

“R_E” refers to the expected return on equity or the cost of equity

“R_f” refers to the risk-free rate

“R_M” refers to the expected return on the market portfolio

“β_E” refers to the beta or risk of the equity

The formula to calculate the cost of preferred stock:

$R_P=\frac{D}{P_0}$

Where,

“R_P” refers to the return on preferred stock or cost of preferred stock

“D” refers to the dividend earned on the preferred stock

“P₀” refers to the current price of preference stock

The formula to calculate annual coupon payment:

$\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}$

The formula to calculate the current price or the market value of the debt:

$\text{Current price}=\text{Face value of the debt}\times \text{Last price percentage}$

The formula to calculate the yield to maturity:

$\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}$

Where,

“C” refers to the coupon paid per period

“F” refers to the face value paid at maturity

“r” refers to the yield to maturity

“t” refers to the periods to maturity

The formula to calculate the after-tax cost of debt:

$\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)$

Where,

“R_D” refers to the cost of debt

“T_C” refers to the corporate tax rate

The formula to calculate the weighted average cost of capital:

$WACC=\left(\frac{E}{V}\right)\times R_E+\left(\frac{P}{V}\right)\times R_P+\left[\left(\frac{D}{V}\right)\times R_D\times \left(1-T_C\right)\right]$

Where,

“WACC” refers to the weighted average cost of capital

“R_E” refers to the return on equity or cost of equity

“R_D” refers to the return on debt or cost of debt

“R_P” refers to the return on debt or cost of preferred stock

“E” refers to the amount of common equity capital

“D” refers to the amount of debt

“P” refers to the amount of preferred stock

“V” refers to the total amount of capital

“T_C” refers to the corporate tax rate

Compute the market value of debt:

$\begin{array}{c}\text{Market value of debt}=\left(\begin{array}{l}\text{Face value}\\ \text{of one bond}\end{array}\right)\times \left(\begin{array}{l}\text{Number of}\\ \text{bonds}\end{array}\right)\times \text{Price quote}\\ =\$1,000\times 40,000\times \frac{95}{100}\\ =\$38,000,000\end{array}$

Hence, the market value of debt is $38,000,000.

Compute the market value of equity:

$\begin{array}{c}\text{Market value of equity }(E)=\text{Outstanding equity shares}\times \text{Market value per share}\\ =825,000\times \$97\\ =\$80,025,000\end{array}$

Hence, the market value of common equity is $80,025,000.

Compute the market value of preferred stock:

$\begin{array}{c}\begin{array}{l}\text{Market value of}\\ \text{preferred shares }(P)\end{array}\}=\text{Outstanding preferred shares}\times \text{Market value per share}\\ =45,000\times \$95\\ =\$4,275,000\end{array}$

Hence, the market value of preferred stock is $4,275,000.

Compute the total market value of the capital structure:

$\begin{array}{c}\text{Total market value }(V)=\left(\begin{array}{l}\text{Market value}\\ \text{of equity }(E)\end{array}\right)+\left(\begin{array}{l}\text{Market value of}\\ \text{preferred stock }(P)\end{array}\right)+\left(\begin{array}{l}\text{Market value}\\ \text{of debt }(D)\end{array}\right)\\ =\$80,025,000+\$4,275,000+\$38,000,000\\ =\$122,300,000\end{array}$

Hence, the total market value of the capital structure is $122,300,000.

Compute the cost of equity:

$\begin{array}{c}{R}_E=R_f+\left[R_M-R_f\right]\times {\beta }_E\\ =0.038+\left[0.07\right]\times 1.15\\ =0.038+0.0805\\ =0.01185\text{ or 11}\text{.85}\%\end{array}$

Hence, the cost of equity is 11.85 percent.

Compute the cost of preferred stock:

The face value of the preferred stock is $100. The dividend percent is 5.8. Hence, the dividend is $5.8 $(\$100\times 5.8\%)$.

$\begin{array}{c}{R}_P=\frac{D}{P_0}\\ =\frac{\$5.8}{\$95}\\ =0.0611\text{ or }6.11\%\end{array}$

Hence, the cost of preferred stock is 6.11 percent.

Compute the annual coupon payment:

$\begin{array}{c}\text{Annual coupon payment}=\text{Face value of the bond}\times \text{Coupon rate}\\ =\$1,000\times 6.2\%\\ =\$62\end{array}$

Hence, the annual coupon payment is $62.

Compute the current price of the bond:

The face value of the bond is $1,000. The bond value is 95% of the face value of the bond.

$\begin{array}{c}\text{Current price}=\text{Face value of the bond}\times \text{Last price percentage}\\ =\$1,000\times \frac{95}{100}\\ =\$950\end{array}$

Hence, the current price of the bond is $950.

Compute the semiannual yield to maturity of the bond as follows:

The bond pays the coupons semiannually. The annual coupon payment is $62. However, the bondholder will receive the same is two equal installments. Hence, semiannual coupon payment or the 6-month coupon payment is $31 $\left(\$62÷2\right)$.

The time to maturity is 25 years. As the coupon payment is semiannual, the semiannual periods to maturity are 50 $\left(25\text{ years}\times 2\right)$. In other words, “t” equals to 50 6-month periods.

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ \$950=\$31\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^{50}}\right]}{r}+\frac{\$1,000}{{\left(1+r\right)}^{50}}\end{array}$ Equation (1)

Finding “r” in Equation (1) would give the semiannual yield to maturity. However, it is difficult to simplify the above the equation. Hence, the only method to solve for “r” is the trial and error method.

The first step in trial and error method is to identify the discount rate that needs to be used. The bond sells at a premium in the market if the market rates (Yield to maturity) are lower than the coupon rate. Similarly, the bond sells at a discount if the market rate (Yield to maturity) is greater than the coupon rate.

In the given information, the bond sells at a discount because the market value of the bond is lower than its face value. Hence, substitute “r” with a rate that is higher than the coupon rate until one obtains the bond value close to $950.

The coupon rate of 6.2 percent is an annual rate. The semiannual coupon rate is 3.1 percent $\left(\text{6}\text{.2 percent}÷2\right)$. The trial rate should be above 3.1 percent.

The attempt under the trial and error method using 3.306 percent as “r”:

$\begin{array}{c}\text{Bond value}=C\times \frac{\left[1-\frac{1}{{\left(1+r\right)}^t}\right]}{r}+\frac{F}{{\left(1+r\right)}^t}\\ =\$31\times \frac{\left[1-\frac{1}{{\left(1+0.03306\right)}^{50}}\right]}{0.03306}+\frac{\$1,000}{{\left(1+0.03306\right)}^{50}}\\ =\$753.28+\$196.66\\ =\$949.94\end{array}$

The current price of the bond is $949.94 when “r” is 3.306 percent. Hence, 3.306 percent is the semiannual yield to maturity.

Compute the annual yield to maturity:

$\begin{array}{c}\text{Yield to maturity}=\text{Semiannual yield to maturity}\times \text{2}\\ =3.306\%\times 2\\ =6.61\%\end{array}$

Hence, the yield to maturity is 6.61 percent.

Compute the after-tax cost of debt:

The pre-tax cost of debt is equal to the yield to maturity of the bond. The yield to maturity of the bond is 6.61 percent. The corporate tax rate is 34 percent.

$\begin{array}{c}\text{After-tax}\text{}{R}_D=R_D\times (1-T_C)\\ =0.0661\times (1-0.34)\\ =0.0661\times 0.66\\ =0.0436\text{ or }4.36\%\end{array}$

Hence, the after-tax cost of debt is 4.36 percent.

Compute the weighted average cost of capital:

The market value of equity (E) is $80,025,000, the market value of preferred stock (P) is $4,275,000, and the market value of debt (D) is $38,000,000. The total market value (V) is $122,300,000. The cost of equity “R_E” is 11.31 percent, the cost of preferred stock “R_P” is 6.11 percent, and the after-tax cost of debt “R_D” is 4.36 percent. As the after-tax cost of debt is available, omit “1‑T_C” from the formula.

$\begin{array}{c}WACC=\left(\frac{E}{V}\right)\times R_E+\left(\frac{P}{V}\right)\times R_P+\left(\frac{D}{V}\right)\times R_D\\ =\left[\begin{array}{l}\left(\frac{\$80,025,000}{\$122,300,000}\times 0.1185\right)+\left(\frac{\$4,275,000}{\$122,300,000}\times 0.0611\right)+\\ \left(\frac{\$38,000,000}{\$122,300,000}\times 0.0436\right)\end{array}\right]\\ =0.0775+0.0021+0.0135\\ =0.0931\text{ or 9}\text{.31}\%\end{array}$

Hence, the weighted average cost of capital is 9.31 percent.

Compute the discount rate:

The company adds 2 percent to its weighted average cost of capital to match the riskiness of the project. The weighted average cost of capital is 6.45 percent. Hence, the discount rate is 11.31 percent $(9.31\%+2\%)$.

c)

Summary Introduction

To determine: The after-tax salvage value after 5 years

Introduction:

Project evaluation refers to the process of evaluating the project for its feasibility and economic viability.

Answer to Problem 25QP

The after-tax salvage value after 5 years is $3,298,500.

Explanation of Solution

The formula to calculate the depreciation per year:

$\text{Depreciation}=\frac{\text{Cost of the equipment}}{\text{Number of years of tax life}}$

The formula to calculate the book value after 5 years:

$\text{Book value after 5 years}=\text{Cost of the equipment}-\left(\begin{array}{l}\text{Depreciation per year}\times \\ 5\text{ years}\end{array}\right)$

The formula to calculate the after-tax salvage value:

$\text{After-tax salvage value}=\text{Salvage value}+\left(\text{Book value}-\text{Market value}\right)\times \text{Tax rate}$

Compute the depreciation per year:

The cost of the equipment is $15,000,000, and its tax life is 8 years (Given).

$\begin{array}{c}\text{Depreciation}=\frac{\text{Cost of the equipment}}{\text{Number of years of tax life}}\\ =\frac{\$15,000,000}{8\text{ years}}\\ =\$1,875,000\end{array}$

Hence, the depreciation per year is $1,875,000.

Compute the book value after 5 years:

$\begin{array}{c}\text{Book value after 5 years}=\text{Cost of the equipment}-\left(\begin{array}{l}\text{Depreciation per year}\times \\ 5\text{ years}\end{array}\right)\\ =\$15,000,000-\left(\$1,875,000\times 5\text{ years}\right)\\ =\$15,000,000-\$9,375,000\\ =\$5,625,000\end{array}$

Hence, the book value after five years is $5,625,000.

Compute the after-tax salvage value:

It is given that the market value of the equipment after 5 years is $2,100,000. The book value after 5 years is $5,625,000. The tax rate is 34 percent.

$\begin{array}{c}\text{After-tax salvage value}=\text{Market value}+\left(\text{Book value}-\text{Market value}\right)\times \text{Tax rate}\\ =\$2,100,000+\left(\$5,625,000-\$2,100,000\right)\times 0.34\\ =\$2,100,000+\$1,198,500\\ =\$3,298,500\end{array}$

Hence, the after-tax salvage value is $3,298,500.

d)

Summary Introduction

To determine: The operating cash flow of the project

Introduction:

Project evaluation refers to the process of evaluating the project for its feasibility and economic viability.

Answer to Problem 25QP

The operating cash flow of the project every year is $5,455,500.

Explanation of Solution

Given information:

The new project has annual fixed costs worth $3,500,000. The new plant will manufacture 12,000 products. The selling price of each product is $10,800, and the cost of production is $9,900.

The formula to calculate the operating cash flow:

$\text{Operating cash flow}=\left(\begin{array}{l}\left[\begin{array}{l}\left(\text{Selling price}–\text{Cost price}\right)\times \\ \text{Number of units produced}–\text{ Fixed costs}\end{array}\right]\times \\ \left(1–\text{Tax rate}\right)\end{array}\right)+\left(\begin{array}{l}\text{Tax rate}\times \\ \text{Depreciation}\end{array}\right)$

Compute the operating cash flow:

The selling price of each product is $10,800, and the cost of production is $9,900. The depreciation per year is $1,875,000. The fixed costs are $3,500,000. The tax rate is 34 percent.

$\begin{array}{c}\text{Operating cash flow}=\left(\begin{array}{l}\left[\begin{array}{l}\left(\text{Selling price}–\text{Cost price}\right)\times \\ \text{Number of units produced}–\text{ Fixed costs}\end{array}\right]\times \\ \left(1–\text{Tax rate}\right)\end{array}\right)+\left(\begin{array}{l}\text{Tax rate}\times \\ \text{Depreciation}\end{array}\right)\\ =\left(\begin{array}{c}\left(\left[\left(\$10,800–\$9,900\right)\times 12,000–\$3,500,000\right]\times \left(1–0.34\right)\right)+\\ \left(0.34\times \$1,875,000\right)\end{array}\right)\\ =\$4,818,000+\$637,500\\ =\$5,455,500\end{array}$

Hence, the operating cash flow is $5,455,500.

e)

Summary Introduction

To determine: The net present value and the internal rate of return

Introduction:

Project evaluation refers to the process of evaluating the project for its feasibility and economic viability.

Answer to Problem 25QP

The net present value is $4,696,660.43, and the internal rate of return is 18.6401270 percent.

Explanation of Solution

Net present value is a tool that helps in evaluating a new project. It determines whether the present value of cash flows from the project exceed the initial cost of investment. The internal rate of return in a return where the net present value of the project is zero.

The formula to calculate the net present value and the internal rate of return:

$\text{Net present value}=\left(\frac{{\text{CF}}_1}{1+r}+\frac{{\text{CF}}_2}{{\left(1+r\right)}^2}+\frac{{\text{CF}}_3}{{\left(1+r\right)}^3}+\mathrm{...}+\frac{{\text{CF}}_t}{{\left(1+r\right)}^t}\right)-\text{Initial investment}$

Where,

“CF₁ to CF_t” refer to the cash flow from period 1 to period t

“r” refers to the discount rate

Compute the net present value:

The initial investment in the project is $20,825,000. The operating cash flow of the project is $5,455,500. Hence, it is the cash flow from Year 1 to Year 4. For Year 5, the cash flow should include the operating cash flow plus the after-tax salvage value, after-tax land value, and the recovery of the initially invested working capital.

The after-tax salvage value is $3,298,500 the after-tax land value is $5,300,000, and the initial working capital is $825,000. Hence, the cash flow in Year 5 is $14,879,000 $\left(\$5,455,500+\$3,298,500+\$5,300,000+\$825,000\right)$. The discount rate is 8.45 percent.

$\begin{array}{c}\text{Net present value}=\left(\frac{{\text{CF}}_1}{1+r}+\frac{{\text{CF}}_2}{{\left(1+r\right)}^2}+\frac{{\text{CF}}_3}{{\left(1+r\right)}^3}+\mathrm{...}+\frac{{\text{CF}}_t}{{\left(1+r\right)}^t}\right)-\text{Initial investment}\\ =\left(\begin{array}{l}\left(\begin{array}{l}\frac{\$5,455,500}{1+0.1131}+\frac{\$5,455,500}{{\left(1+0.1131\right)}^2}+\frac{\$5,455,500}{{\left(1+0.1131\right)}^3}+\\ \frac{\$5,455,500}{{\left(1+0.1131\right)}^4}+\frac{\$14,879,000}{{\left(1+0.1131\right)}^5}\end{array}\right)\\ -\$20,825,000\end{array}\right)\\ =\left(\begin{array}{l}\$4,901,176.893+\$4,403,177.516+\\ \$3,955,778.92+\$3,553,839.655+\\ \$8,707,687.405\end{array}\right)-\$20,825,000\\ =\$4,696,660.43\end{array}$

Hence, the net present value is $4,696,660.43.

Compute the internal rate of return:

The internal rate of return in a return where the net present value of the project is zero. It is possible to determine the internal rate of return only through the trail and error method or by using a software. The net present value is positive. Hence, the trial and error rate should be higher than the discount rate. Use a trial rate of 18.6401270 percent to determine whether the net present value is zero.

$\begin{array}{c}\text{Net present value}=\left(\frac{{\text{CF}}_1}{1+r}+\frac{{\text{CF}}_2}{{\left(1+r\right)}^2}+\frac{{\text{CF}}_3}{{\left(1+r\right)}^3}+\mathrm{...}+\frac{{\text{CF}}_t}{{\left(1+r\right)}^t}\right)-\text{Initial investment}\\ 0=\left(\begin{array}{l}\left(\begin{array}{l}\frac{\$5,455,500}{1+r}+\frac{\$5,455,500}{{\left(1+r\right)}^2}+\frac{\$5,455,500}{{\left(1+r\right)}^3}+\\ \frac{\$5,455,500}{{\left(1+r\right)}^4}+\frac{\$5,455,500}{{\left(1+r\right)}^5}\end{array}\right)\\ -\$20,825,000\end{array}\right)\\ 0=\left(\begin{array}{l}\left(\begin{array}{l}\frac{\$5,455,500}{1+0.186401270}+\frac{\$5,455,500}{{\left(1+0.186401270\right)}^2}+\frac{\$5,455,500}{{\left(1+0.186401270\right)}^3}+\\ \frac{\$5,455,500}{{\left(1+0.186401270\right)}^4}+\frac{\$14,879,000}{{\left(1+0.186401270\right)}^5}\end{array}\right)\\ -\$20,825,500\end{array}\right)\\ 0=\$20,825,000-\$20,825,000\\ 0=\$0\end{array}$

Hence, the approximate internal rate of return is 18.6401270 percent.