Problem 1: Relevant Cash Flows
Parker & Stone, Inc., is looking at setting up a new manufacturing plant in South Park to produce garden tools. The company bought some land six years ago for $2.8 million in anticipation of using it as a warehouse and distribution site, but the company has since decided to rent these facilities from a competitor instead. If the land were sold today, the company would net $3.2 million. The company wants to build its new manufacturing plant on this land; the plant will cost $14.3 million to build, and the site requires $825,000 worth of grading before it is suitable for construction. What is the proper cash flow amount to use as the initial investment in fixed assets when evaluating this project? Why?
Solution
This problem tests the concepts of sunk costs and opportunity costs (Section 10.2).
Step 1: Identify Relevant Cash Flows.
- The initial cost of the land ($2.8 million) is a sunk cost. It has already been paid and will not change regardless of the current decision. Therefore, it is irrelevant.
- The after-tax proceeds from a potential sale of the land today ($3.2 million) represent an opportunity cost. By using the land for the new plant, the company gives up the opportunity to receive this cash inflow. This is a relevant cash flow.
- The cost to build the plant ($14.3 million) is a new cash outflow directly caused by the project. This is a relevant cash flow.
- The cost of grading ($825,000) is also a new, direct cash outflow. This is a relevant cash flow.
Step 2: Calculate the Total Initial Investment.
$$\text{Initial Investment} = \text{Opportunity Cost} + \text{Build Cost} + \text{Grading Cost}$$
$$\text{Initial Investment} = \$3,200,000 + \$14,300,000 + \$825,000 = \textbf{\$18,325,000}$$
Problem 2: Erosion (Side Effects)
Winnebagel Corp. currently sells 20,000 motor homes per year at $103,000 each and 14,000 luxury motor coaches per year at $155,000 each. The company wants to introduce a new portable camper to fill out its product line; it hopes to sell 25,000 of these campers per year at $19,000 each. An independent consultant has determined that if the company introduces the new campers, it should boost the sales of its existing motor homes by 2,700 units per year and reduce the sales of its motor coaches by 1,300 units per year. What is the amount to use as the annual sales figure when evaluating this project? Why?
Solution
This problem addresses the concept of side effects, specifically synergy (positive) and erosion (negative), as discussed in Section 10.2.
Step 1: Calculate the incremental sales for each product line.
- New Campers: The new sales figure is the direct result of the project.
$$25,000 \text{ units} \times \$19,000/\text{unit} = \$475,000,000$$ - Motor Coaches (Erosion): Sales are lost due to the new project. This is a negative incremental cash flow.
$$1,300 \text{ units} \times \$155,000/\text{unit} = \$201,500,000$$ - Motor Homes (Synergy): Sales are boosted by the new project. This is a positive incremental cash flow.
$$2,700 \text{ units} \times \$103,000/\text{unit} = \$278,100,000$$
Step 2: Calculate the total incremental annual sales figure.
$$\text{Incremental Sales} = \text{New Sales} - \text{Lost Sales} + \text{Boosted Sales}$$
$$\text{Incremental Sales} = \$475,000,000 - \$201,500,000 + \$278,100,000 = \textbf{\$551,600,000}$$
Problem 3: Calculating Projected Net Income
A proposed new investment has projected sales of $515,000. Variable costs are 36 percent of sales, and fixed costs are $173,000; depreciation is $46,000. Prepare a pro forma income statement assuming a tax rate of 21 percent. What is the projected net income?
Solution
This is a straightforward preparation of a pro forma income statement, as covered in Section 10.3.
Step 1: Calculate variable costs.
$$\text{Variable Costs} = 0.36 \times \text{Sales} = 0.36 \times \$515,000 = \$185,400$$
Step 2: Calculate EBIT (Earnings Before Interest and Taxes).
$$\text{EBIT} = \text{Sales} - \text{Variable Costs} - \text{Fixed Costs} - \text{Depreciation}$$
$$\text{EBIT} = \$515,000 - \$185,400 - \$173,000 - \$46,000 = \$110,600$$
Step 3: Calculate Taxes.
$$\text{Taxes} = \text{EBIT} \times \text{Tax Rate} = \$110,600 \times 0.21 = \$23,226$$
Step 4: Calculate Net Income.
$$\text{Net Income} = \text{EBIT} - \text{Taxes} = \$110,600 - \$23,226 = \textbf{\$87,374}$$
Pro Forma Income Statement:
| Description | Amount |
|---|---|
| Sales | $515,000 |
| Variable costs | $185,400 |
| Fixed costs | $173,000 |
| Depreciation | $46,000 |
| EBIT | $110,600 |
| Taxes (21%) | $23,226 |
| Net Income | $87,374 |
Problem 4: Calculating OCF
Consider the following income statement:
| Description | Amount |
|---|---|
| Sales | $704,600 |
| Costs | 527,300 |
| Depreciation | 82,100 |
| EBIT | ? |
| Taxes (22%) | ? |
| Net Income | ? |
Solution
This problem reinforces the relationship between income statement components and Operating Cash Flow (OCF), a key concept from Section 10.3.
Step 1: Fill in the missing numbers on the income statement.
$$\text{EBIT} = \text{Sales} - \text{Costs} - \text{Depreciation}$$
$$\text{EBIT} = \$704,600 - \$527,300 - \$82,100 = \textbf{\$95,200}$$
$$\text{Taxes} = \text{EBIT} \times \text{Tax Rate} = \$95,200 \times 0.22 = \textbf{\$20,944}$$
$$\text{Net Income} = \text{EBIT} - \text{Taxes} = \$95,200 - \$20,944 = \textbf{\$74,256}$$
Step 2: Calculate OCF.
$$\text{OCF} = \text{EBIT} + \text{Depreciation} - \text{Taxes}$$
$$\text{OCF} = \$95,200 + \$82,100 - \$20,944 = \textbf{\$156,356}$$
Step 3: Calculate the Depreciation Tax Shield.
$$\text{Depreciation Tax Shield} = \text{Depreciation} \times \text{Tax Rate}$$
$$\text{Depreciation Tax Shield} = \$82,100 \times 0.22 = \textbf{\$18,062}$$
Problem 5: OCF from Several Approaches
A proposed new project has projected sales of $215,000, costs of $104,000, and depreciation of $25,300. The tax rate is 23 percent. Calculate operating cash flow using the four different approaches described in the chapter and verify that the answer is the same in each case.
Solution
This problem demonstrates the equivalence of the four OCF calculation methods introduced in Section 10.5. First, we need to calculate EBIT, Taxes, and Net Income.
$$\text{EBIT} = \text{Sales} - \text{Costs} - \text{Depreciation} = \$215,000 - \$104,000 - \$25,300 = \$85,700$$
$$\text{Taxes} = \text{EBIT} \times \text{Tax Rate} = \$85,700 \times 0.23 = \$19,711$$
$$\text{Net Income} = \text{EBIT} - \text{Taxes} = \$85,700 - \$19,711 = \$65,989$$
Method 1: Standard Approach
$$\text{OCF} = \text{EBIT} + \text{Depreciation} - \text{Taxes}$$
$$\text{OCF} = \$85,700 + \$25,300 - \$19,711 = \textbf{\$91,289}$$
Method 2: Bottom-Up Approach
$$\text{OCF} = \text{Net Income} + \text{Depreciation}$$
$$\text{OCF} = \$65,989 + \$25,300 = \textbf{\$91,289}$$
Method 3: Top-Down Approach
$$\text{OCF} = \text{Sales} - \text{Costs} - \text{Taxes}$$
$$\text{OCF} = \$215,000 - \$104,000 - \$19,711 = \textbf{\$91,289}$$
Method 4: Tax Shield Approach
$$\text{OCF} = (\text{Sales} - \text{Costs}) \times (1 - T_C) + \text{Depreciation} \times T_C$$
$$\text{OCF} = (\$215,000 - \$104,000) \times (1 - 0.23) + (\$25,300 \times 0.23)$$
$$\text{OCF} = (\$111,000 \times 0.77) + \$5,819 = \$85,470 + \$5,819 = \textbf{\$91,289}$$
Key Insight
All four methods produce the same result because they are mathematically equivalent. The best approach to use is often the one where you have the most readily available information.
Problem 6: Calculating Depreciation
A piece of newly purchased industrial equipment costs $1.475 million and is classified as seven-year property under MACRS. Calculate the annual depreciation allowances and end-of-the-year book values for this equipment.
Solution
This problem applies the MACRS depreciation schedule from Section 10.4. We use the percentages for a seven-year property class to calculate the annual depreciation and remaining book value.
MACRS Depreciation and Book Value:
| Year | MACRS % | Depreciation | Ending Book Value |
|---|---|---|---|
| 1 | 14.29% | \(1,475,000 \times 0.1429 = \$210,757.50\) | \(\$1,475,000 - \$210,757.50 = \$1,264,242.50\) |
| 2 | 24.49% | \(1,475,000 \times 0.2449 = \$361,202.50\) | \(\$1,264,242.50 - \$361,202.50 = \$903,040.00\) |
| 3 | 17.49% | \(1,475,000 \times 0.1749 = \$257,977.50\) | \(\$903,040.00 - \$257,977.50 = \$645,062.50\) |
| 4 | 12.49% | \(1,475,000 \times 0.1249 = \$184,287.50\) | \(\$645,062.50 - \$184,287.50 = \$460,775.00\) |
| 5 | 8.93% | \(1,475,000 \times 0.0893 = \$131,667.50\) | \(\$460,775.00 - \$131,667.50 = \$329,107.50\) |
| 6 | 8.92% | \(1,475,000 \times 0.0892 = \$131,520.00\) | \(\$329,107.50 - \$131,520.00 = \$197,587.50\) |
| 7 | 8.93% | \(1,475,000 \times 0.0893 = \$131,667.50\) | \(\$197,587.50 - \$131,667.50 = \$65,920.00\) |
| 8 | 4.46% | \(1,475,000 \times 0.0446 = \$65,885.00\) | \(\$65,920.00 - \$65,885.00 = \$35.00\) |
Problem 7: Calculating Salvage Value (Straight-Line)
Consider an asset that costs $745,000 and is depreciated straight-line to zero over its eight-year tax life. The asset is to be used in a five-year project; at the end of the project, the asset can be sold for $135,000. If the relevant tax rate is 21 percent, what is the aftertax cash flow from the sale of this asset?
Solution
This problem requires us to calculate the after-tax cash flow from a sale, accounting for taxes on a loss (Section 10.4).
Step 1: Calculate the annual depreciation and book value at the end of the project's life.
$$\text{Annual Depreciation} = \frac{\$745,000}{8 \text{ years}} = \$93,125$$
$$\text{Total Depreciation (after 5 years)} = \$93,125 \times 5 = \$465,625$$
$$\text{Book Value (end of Year 5)} = \text{Initial Cost} - \text{Total Depreciation}$$
$$\text{Book Value} = \$745,000 - \$465,625 = \textbf{\$279,375}$$
Step 2: Determine the taxable gain or loss.
The asset is sold for $135,000, which is less than its book value of $279,375. This results in a taxable loss.
$$\text{Taxable Loss} = \text{Book Value} - \text{Salvage Value}$$
$$\text{Taxable Loss} = \$279,375 - \$135,000 = \$144,375$$
Step 3: Calculate the after-tax cash flow from the sale.
The tax loss generates a tax shield, which is a cash inflow from tax savings.
$$\text{Tax Savings} = \text{Taxable Loss} \times \text{Tax Rate} = \$144,375 \times 0.21 = \textbf{\$30,318.75}$$
$$\text{After-Tax Cash Flow} = \text{Salvage Value} + \text{Tax Savings}$$
$$\text{After-Tax Cash Flow} = \$135,000 + \$30,318.75 = \textbf{\$165,318.75}$$
Problem 8: Calculating Salvage Value (MACRS)
An asset used in a four-year project falls in the five-year MACRS class for tax purposes. The asset has an acquisition cost of $5.7 million and will be sold for $1.8 million at the end of the project. If the tax rate is 21 percent, what is the aftertax salvage value of the asset?
Solution
This problem is similar to the previous one, but requires the use of MACRS percentages to determine the book value (Section 10.4).
Step 1: Calculate the book value at the end of Year 4.
Using the 5-year MACRS percentages (20.00%, 32.00%, 19.20%, 11.52%) from the textbook, we find the total accumulated depreciation after four years.
$$\text{Total Depreciation} = (\$5.7\text{M} \times 0.20) + (\$5.7\text{M} \times 0.32) + (\$5.7\text{M} \times 0.1920) + (\$5.7\text{M} \times 0.1152)$$
$$\text{Total Depreciation} = \$1.14\text{M} + \$1.824\text{M} + \$1.0944\text{M} + \$0.65664\text{M} = \$4,715,040$$
$$\text{Book Value (end of Year 4)} = \$5.7\text{M} - \$4,715,040 = \textbf{\$984,960}$$
Step 2: Calculate the after-tax salvage value.
The salvage value ($1.8 million) is greater than the book value ($984,960), resulting in a taxable gain.
$$\text{Taxable Gain} = \text{Salvage Value} - \text{Book Value}$$
$$\text{Taxable Gain} = \$1,800,000 - \$984,960 = \$815,040$$
$$\text{Taxes on Gain} = \$815,040 \times 0.21 = \$171,158.40$$
$$\text{After-Tax Salvage Value} = \text{Salvage Value} - \text{Taxes on Gain}$$
$$\text{After-Tax Salvage Value} = \$1,800,000 - \$171,158.40 = \textbf{\$1,628,841.60}$$
Problem 9: Calculating Project OCF
Esfandairi Enterprises is considering a new three-year expansion project that requires an initial fixed asset investment of $2.18 million. The fixed asset will be depreciated straight-line to zero over its three-year tax life, after which time it will be worthless. The project is estimated to generate $1.645 million in annual sales, with costs of $610,000. If the tax rate is 21 percent, what is the OCF for this project?
Solution
This problem is a direct application of the OCF formula, as outlined in Section 10.3 and 10.5.
Step 1: Calculate the annual depreciation.
$$\text{Annual Depreciation} = \frac{\$2,180,000}{3} = \$726,666.67$$
Step 2: Calculate the annual OCF.
We will use the tax shield approach for clarity.
$$\text{OCF} = (\text{Sales} - \text{Costs}) \times (1 - T_C) + \text{Depreciation} \times T_C$$
$$\text{OCF} = (\$1,645,000 - \$610,000) \times (1 - 0.21) + (\$726,666.67 \times 0.21)$$
$$\text{OCF} = (\$1,035,000 \times 0.79) + \$152,600 = \$817,650 + \$152,600 = \textbf{\$970,250}$$
Problem 10: Calculating Project NPV
In the previous problem, suppose the required return on the project is 12 percent. What is the project's NPV?
Solution
This problem builds on Problem 9 by calculating the Net Present Value (NPV) using the cash flows determined previously (Section 10.3).
Step 1: Identify the cash flows.
- Year 0: Initial fixed asset investment = -$2,180,000
- Years 1-3: OCF = +$970,250
Step 2: Calculate NPV.
$$NPV = -\$2,180,000 + \frac{\$970,250}{(1.12)^1} + \frac{\$970,250}{(1.12)^2} + \frac{\$970,250}{(1.12)^3}$$
$$NPV = -\$2,180,000 + \$866,294.64 + \$773,477.36 + \$690,604.79 = \textbf{\$150,376.79}$$
Interpretation
Since the NPV is positive, the project is considered profitable and should be accepted.
Problem 11: Calculating Project Cash Flow from Assets
In the previous problem, suppose the project requires an initial investment in net working capital of $250,000, and the fixed asset will have a market value of $180,000 at the end of the project. What is the project's Year 0 net cash flow? Year 1? Year 2? Year 3? What is the new NPV?
Solution
This problem adds Net Working Capital (NWC) and a salvage value to the cash flow analysis (Sections 10.2 and 10.4).
Step 1: Identify all cash flow components.
- Initial Investment: Fixed asset of -$2,180,000 and NWC of -$250,000.
- OCF: Remains the same at +$970,250 per year (from Problem 9).
- Terminal Cash Flow (Year 3): * NWC recovery: +$250,000 * After-tax salvage value: Book value is $0 (straight-line to zero). The salvage value of $180,000 is a taxable gain. * Tax on gain: $180,000 \times 0.21 = \$37,800$ * After-tax salvage: $180,000 - \$37,800 = \$142,200$
Step 2: Summarize total cash flows by year.
| Cash Flow Component | Year 0 | Year 1 | Year 2 | Year 3 |
|---|---|---|---|---|
| OCF | +$970,250 | +$970,250 | +$970,250 | |
| NWC | -$250,000 | +$250,000 | ||
| Capital Spending | -$2,180,000 | +$142,200 | ||
| Total Cash Flow | -$2,430,000 | $970,250 | $970,250 | $1,362,450 |
Step 3: Calculate the new NPV.
$$NPV = -\$2,430,000 + \frac{\$970,250}{(1.12)^1} + \frac{\$970,250}{(1.12)^2} + \frac{\$1,362,450}{(1.12)^3}$$
$$NPV = -\$2,430,000 + \$866,294.64 + \$773,477.36 + \$969,720.48 = \textbf{\$179,492.48}$$
Problem 12: NPV and MACRS
In the previous problem, suppose the fixed asset actually falls into the three-year MACRS class. All the other facts are the same. What is the project's Year 1 net cash flow now? Year 2? Year 3? What is the new NPV?
Solution
This problem shows the impact of an accelerated depreciation schedule on cash flows and NPV (Section 10.4).
Step 1: Calculate annual MACRS depreciation and book value.
- Year 1: \(2.18\text{M} \times 0.3333 = \$726,594\). Book Value: \(2.18\text{M} - 726,594 = \$1,453,406\)
- Year 2: \(2.18\text{M} \times 0.4445 = \$968,810\). Book Value: \(1,453,406 - 968,810 = \$484,596\)
- Year 3: \(2.18\text{M} \times 0.1481 = \$322,878\). Book Value: \(484,596 - 322,878 = \$161,718\)
Step 2: Calculate OCF for each year (using the tax shield approach).
Note: The after-tax sales minus costs component is constant at \((\$1,645,000 - \$610,000) \times 0.79 = \$817,650\).
$$\text{OCF}_1 = \$817,650 + (\$726,594 \times 0.21) = \$817,650 + \$152,584.74 = \textbf{\$970,234.74}$$
$$\text{OCF}_2 = \$817,650 + (\$968,810 \times 0.21) = \$817,650 + \$203,450.10 = \textbf{\$1,021,100.10}$$
$$\text{OCF}_3 = \$817,650 + (\$322,878 \times 0.21) = \$817,650 + \$67,804.38 = \textbf{\$885,454.38}$$
Step 3: Calculate the after-tax salvage value.
$$\text{Taxable Gain} = \text{Salvage} - \text{Book Value} = \$180,000 - \$161,718 = \$18,282$$
$$\text{After-Tax Salvage} = \$180,000 - (\$18,282 \times 0.21) = \$180,000 - \$3,839.22 = \textbf{\$176,160.78}$$
Step 4: Summarize total cash flows and calculate NPV.
| Cash Flow Component | Year 0 | Year 1 | Year 2 | Year 3 |
|---|---|---|---|---|
| OCF | +$970,234.74 | +$1,021,100.10 | +$885,454.38 | |
| NWC | -$250,000 | +$250,000 | ||
| Capital Spending | -$2,180,000 | +$176,160.78 | ||
| Total Cash Flow | -$2,430,000 | $970,234.74 | $1,021,100.10 | $1,311,615.16 |
$$NPV = -\$2,430,000 + \frac{\$970,234.74}{(1.12)^1} + \frac{\$1,021,100.10}{(1.12)^2} + \frac{\$1,311,615.16}{(1.12)^3} = \textbf{\$194,541.40}$$
Problem 13: NPV and Bonus Depreciation
In the previous problem, suppose the fixed asset actually qualifies for 100 percent bonus depreciation in the first year. All the other facts are the same. What is the project's Year 1 net cash flow now? Year 2? Year 3? What is the new NPV?
Solution
Bonus depreciation allows for the entire asset cost to be depreciated in the first year, creating a massive tax shield. This significantly alters the cash flow timeline (Section 10.4).
Step 1: Calculate annual depreciation.
$$\text{Year 1 Depreciation} = \$2,180,000$$
$$\text{Year 2 & 3 Depreciation} = \$0$$
Step 2: Calculate OCF for each year.
$$\text{OCF}_1 = \$817,650 + (\$2,180,000 \times 0.21) = \$817,650 + \$457,800 = \textbf{\$1,275,450}$$
$$\text{OCF}_2 = \$817,650 + (\$0 \times 0.21) = \textbf{\$817,650}$$
$$\text{OCF}_3 = \$817,650 + (\$0 \times 0.21) = \textbf{\$817,650}$$
Step 3: Calculate the after-tax salvage value.
After Year 1, the book value is $0. The taxable gain is the full salvage value of $180,000.
$$\text{Tax on Gain} = \$180,000 \times 0.21 = \$37,800$$
$$\text{After-Tax Salvage} = \$180,000 - \$37,800 = \textbf{\$142,200}$$
Step 4: Summarize total cash flows and calculate NPV.
| Cash Flow Component | Year 0 | Year 1 | Year 2 | Year 3 |
|---|---|---|---|---|
| OCF | +$1,275,450 | +$817,650 | +$817,650 | |
| NWC | -$250,000 | +$250,000 | ||
| Capital Spending | -$2,180,000 | +$142,200 | ||
| Total Cash Flow | -$2,430,000 | $1,275,450 | $817,650 | $1,209,850 |
$$NPV = -\$2,430,000 + \frac{\$1,275,450}{(1.12)^1} + \frac{\$817,650}{(1.12)^2} + \frac{\$1,209,850}{(1.12)^3} = \textbf{\$306,629.39}$$
Problem 14: Project Evaluation (NPV)
Dog Up! Franks is looking at a new sausage system with an installed cost of $385,000. This cost will be depreciated straight-line to zero over the project's five-year life, at the end of which the sausage system can be scrapped for $60,000. The sausage system will save the firm $135,000 per year in pretax operating costs, and the system requires an initial investment in net working capital of $35,000. If the tax rate is 21 percent and the discount rate is 10 percent, what is the NPV of this project?
Solution
This is a complete NPV analysis problem, combining initial investment, OCF, and terminal cash flows (Sections 10.3 and 10.4).
Step 1: Calculate the OCF from cost savings.
$$\text{Annual Depreciation} = \frac{\$385,000}{5} = \$77,000$$
$$\text{OCF} = (\text{Pretax Savings}) \times (1 - T_C) + \text{Depreciation} \times T_C$$
$$\text{OCF} = (\$135,000 \times 0.79) + (\$77,000 \times 0.21) = \$106,650 + \$16,170 = \textbf{\$122,820}$$
Step 2: Calculate the terminal cash flow.
$$\text{Book Value} = 0$$
$$\text{Taxable Gain} = \text{Salvage Value} - \text{Book Value} = \$60,000 - 0 = \$60,000$$
$$\text{Taxes on Gain} = \$60,000 \times 0.21 = \$12,600$$
$$\text{After-Tax Salvage} = \$60,000 - \$12,600 = \textbf{\$47,400}$$
Step 3: Summarize total cash flows.
| Cash Flow Component | Year 0 | Years 1-4 | Year 5 |
|---|---|---|---|
| Initial Investment | -$385,000 | ||
| NWC | -$35,000 | +$35,000 | |
| OCF | +$122,820 | +$122,820 | |
| After-Tax Salvage | +$47,400 | ||
| Total Cash Flow | -$420,000 | $122,820 | $205,220 |
Step 4: Calculate NPV.
$$NPV = -\$420,000 + \sum_{t=1}^{4} \frac{\$122,820}{(1.10)^t} + \frac{\$205,220}{(1.10)^5}$$
$$NPV = -\$420,000 + \$122,820 \times 3.16987 + \frac{\$205,220}{1.61051}$$
$$NPV = -\$420,000 + \$389,013.1 + \$127,425.5 = \textbf{\$96,438.6}$$
Problem 15: NPV and Bonus Depreciation
In the previous problem, suppose the fixed asset actually qualifies for 100 percent bonus depreciation in the first year. What is the new NPV?
Solution
This problem is a direct extension of Problem 14, applying the concept of 100% bonus depreciation (Section 10.4).
Step 1: Calculate the new OCFs.
$$\text{OCF}_1 = (\$135,000 \times 0.79) + (\$385,000 \times 0.21) = \$106,650 + \$80,850 = \textbf{\$187,500}$$
$$\text{OCF}_{2-5} = (\$135,000 \times 0.79) + (\$0 \times 0.21) = \textbf{\$106,650}$$
Step 2: Summarize total cash flows.
| Cash Flow Component | Year 0 | Year 1 | Years 2-4 | Year 5 |
|---|---|---|---|---|
| Initial Investment | -$385,000 | |||
| NWC | -$35,000 | +$35,000 | ||
| OCF | +$187,500 | +$106,650 | +$106,650 | |
| After-Tax Salvage | +$47,400 | |||
| Total Cash Flow | -$420,000 | $187,500 | $106,650 | $189,050 |
Step 3: Calculate the new NPV.
$$NPV = -\$420,000 + \frac{\$187,500}{1.10^1} + \frac{\$106,650}{1.10^2} + \frac{\$106,650}{1.10^3} + \frac{\$106,650}{1.10^4} + \frac{\$189,050}{1.10^5}$$
$$NPV = -\$420,000 + \$170,454.55 + \$88,140.50 + \$80,127.73 + \$72,843.39 + \$117,383.08 = \textbf{\$108,949.25}$$
Key Insight
Bonus depreciation significantly increases the NPV. By accelerating the tax shield, it brings cash flows to the present, which is valuable due to the time value of money.
Problem 16: Project Evaluation (IRR)
Your firm is contemplating the purchase of a new $535,000 computer-based order entry system. The system will be depreciated straight-line to zero over its five-year life. It will be worth $30,000 at the end of that time. You will save $165,000 before taxes per year in order processing costs, and you will be able to reduce working capital by $60,000 (this is a one-time reduction). If the tax rate is 24 percent, what is the IRR for this project?
Solution
This problem requires us to calculate the Internal Rate of Return (IRR) by first determining the project's cash flows (Sections 10.3 and 10.4).
Step 1: Calculate the OCF.
$$\text{Annual Depreciation} = \frac{\$535,000}{5} = \$107,000$$
$$\text{OCF} = (\$165,000 \times (1 - 0.24)) + (\$107,000 \times 0.24)$$
$$\text{OCF} = \$125,400 + \$25,680 = \textbf{\$151,080}$$
Step 2: Calculate the total cash flows.
- Year 0: Initial investment = -$535,000. NWC reduction = +$60,000. Total Year 0 CF = -$475,000.
- Years 1-4: OCF = $151,080.
- Year 5 (Terminal Year): * OCF: +$151,080 * NWC recovery: -$60,000 (cash outflow, as it was freed up initially) * After-tax salvage: Book Value is $0. Taxable gain is $30,000. Tax is $30,000 x 0.24 = $7,200. After-tax salvage is $30,000 - $7,200 = $22,800. * Total Year 5 CF = $151,080 - \$60,000 + \$22,800 = \textbf{\$113,880}
| Year | 0 | 1-4 | 5 |
|---|---|---|---|
| Total Cash Flow | -$475,000 | +$151,080 | +$113,880 |
Step 3: Calculate the IRR.
We solve for the discount rate \(r\) that makes the NPV equal to zero.
$$0 = -\$475,000 + \sum_{t=1}^{4} \frac{\$151,080}{(1+r)^t} + \frac{\$113,880}{(1+r)^5}$$
Using a financial calculator or spreadsheet, the IRR is found to be approximately 17.29%.
Problem 17: Project Evaluation (Sensitivity)
In the previous problem, suppose your required return on the project is 11 percent and your pretax cost savings are $150,000 per year. Will you accept the project? What if the pretax cost savings are $100,000 per year? At what level of pretax cost savings would you be indifferent between accepting the project and not accepting it?
Solution
This problem analyzes the sensitivity of NPV to changes in a key variable: pretax cost savings. This is a form of "what-if" analysis (Section 10.3).
Case 1: Pretax cost savings of $150,000.
First, calculate the new OCF and Year 5 cash flow. Depreciation and NWC effects are unchanged from Problem 16.
$$\text{OCF} = (\$150,000 \times 0.76) + (\$107,000 \times 0.24) = \$114,000 + \$25,680 = \textbf{\$139,680}$$
$$\text{Year 5 CF} = \$139,680 - \$60,000 + \$22,800 = \textbf{\$102,480}$$
Now, calculate the NPV at a required return of 11%.
$$NPV = -\$475,000 + \sum_{t=1}^{4} \frac{\$139,680}{(1.11)^t} + \frac{\$102,480}{(1.11)^5}$$
$$NPV = -\$475,000 + (\$139,680 \times 3.1024) + \frac{\$102,480}{1.685058} = -\$475,000 + \$433,307 + \$60,817 = \textbf{\$19,124}$$
Since the NPV is positive, you should accept the project.
Case 2: Pretax cost savings of $100,000.
$$\text{OCF} = (\$100,000 \times 0.76) + (\$107,000 \times 0.24) = \$76,000 + \$25,680 = \textbf{\$101,680}$$
$$\text{Year 5 CF} = \$101,680 - \$60,000 + \$22,800 = \textbf{\$64,480}$$
Now, calculate the NPV at 11%.
$$NPV = -\$475,000 + \sum_{t=1}^{4} \frac{\$101,680}{(1.11)^t} + \frac{\$64,480}{(1.11)^5}$$
$$NPV = -\$475,000 + (\$101,680 \times 3.1024) + \frac{\$64,480}{1.685058} = -\$475,000 + \$315,441 + \$38,266 = \textbf{-\$121,293}$$
Since the NPV is negative, you should not accept the project.
Case 3: Break-even pretax cost savings.
The break-even point is where NPV equals zero. Let S be the pretax savings.
$$\text{OCF} = S \times (1 - 0.24) + (\$107,000 \times 0.24) = 0.76S + \$25,680$$
$$\text{Year 5 CF} = (0.76S + \$25,680) - \$60,000 + \$22,800 = 0.76S - \$11,520$$
Set NPV = 0 and solve for S.
$$0 = -\$475,000 + (0.76S + \$25,680)\times 3.1024 + \frac{0.76S - \$11,520}{1.685058}$$
$$\$475,000 = (2.3578S + \$79,679) + (0.4510S - \$6,836)$$
$$\$475,000 - \$79,679 + \$6,836 = 2.8088S$$
$$\$402,157 = 2.8088S \implies S = \textbf{\$143,186}$$
Interpretation
You would be indifferent to accepting or rejecting the project at an annual pretax cost savings of $143,186. Below this amount, the project's NPV is negative.
Problem 18: Calculating EAC
A five-year project has an initial fixed asset investment of $345,000, an initial NWC investment of $25,000, and an annual OCF of $41,000. The fixed asset is fully depreciated over the life of the project and has no salvage value. If the required return is 11 percent, what is this project's equivalent annual cost, or EAC?
Solution
This problem demonstrates the calculation of Equivalent Annual Cost (EAC), which is a crucial tool for comparing projects with unequal lives (Section 10.6).
Step 1: Calculate the total PV of all project costs and investments.
The total initial investment is the sum of the fixed asset investment and the NWC investment. The NWC investment is recovered at the end of the project.
$$\text{Total Initial Investment} = \$345,000 + \$25,000 = \$370,000$$
$$\text{Total Cash Flow at Year 5} = \text{OCF} + \text{NWC Recovery} = \$41,000 + \$25,000 = \$66,000$$
We first find the NPV of the project as a whole.
$$NPV = -\$370,000 + \sum_{t=1}^{4} \frac{\$41,000}{(1.11)^t} + \frac{\$66,000}{(1.11)^5}$$
$$NPV = -\$370,000 + (\$41,000 \times 3.1024) + (\$66,000 \times 0.59345)$$
$$NPV = -\$370,000 + \$127,200 + \$39,168 = -\$203,632$$
Step 2: Annualize the NPV to find the EAC.
We find the 5-year annuity factor at 11%.
$$\text{Annuity Factor} = \frac{1 - (1.11)^{-5}}{0.11} = 3.6959$$
$$\text{EAC} = \frac{NPV}{\text{Annuity Factor}} = \frac{-\$203,632}{3.6959} = \textbf{-\$55,108.38}$$
Problem 19: Comparing Two Projects with EAC
You are evaluating two different silicon wafer milling machines. The Techron I costs $265,000, has a three-year life, and has pretax operating costs of $74,000 per year. The Techron II costs $445,000, has a five-year life, and has pretax operating costs of $47,000 per year. For both milling machines, use straight-line depreciation to zero over the project's life and assume a salvage value of $35,000. If your tax rate is 22 percent and your discount rate is 10 percent, compute the EAC for both machines. Which do you prefer? Why?
Solution
This is a classic EAC problem, used to compare mutually exclusive projects with different lives (Section 10.6).
Techron I (3-year life):
Step 1: Calculate annual depreciation and OCF.
$$\text{Depreciation} = \frac{\$265,000 - \$35,000}{3} = \$76,666.67$$
$$\text{OCF} = (-\$74,000 \times 0.78) + (\$76,666.67 \times 0.22) = -\$57,720 + \$16,866.67 = -\$40,853.33$$
Step 2: Calculate after-tax salvage and PV of all costs.
$$\text{After-Tax Salvage} = \$35,000 - (\$35,000 - 0) \times 0.22 = \$27,300$$
$$PV_{costs} = -\$265,000 + \frac{-\$40,853.33}{(1.10)^1} + \frac{-\$40,853.33}{(1.10)^2} + \frac{-\$40,853.33 + \$27,300}{(1.10)^3}$$
$$PV_{costs} = -\$265,000 - \$37,139.39 - \$33,763.08 - \$10,210.36 = -\$346,112.83$$
Step 3: Calculate EAC.
$$\text{3-year Annuity Factor at 10\%} = \frac{1 - (1.10)^{-3}}{0.10} = 2.4869$$
$$\text{EAC} = \frac{-\$346,112.83}{2.4869} = \textbf{-\$139,178.50}$$
Techron II (5-year life):
Step 1: Calculate annual depreciation and OCF.
$$\text{Depreciation} = \frac{\$445,000 - \$35,000}{5} = \$82,000$$
$$\text{OCF} = (-\$47,000 \times 0.78) + (\$82,000 \times 0.22) = -\$36,660 + \$18,040 = -\$18,620$$
Step 2: Calculate after-tax salvage and PV of all costs.
$$\text{After-Tax Salvage} = \$35,000 - (\$35,000 - 0) \times 0.22 = \$27,300$$
$$PV_{costs} = -\$445,000 + \sum_{t=1}^{4} \frac{-\$18,620}{(1.10)^t} + \frac{-\$18,620 + \$27,300}{(1.10)^5}$$
$$PV_{costs} = -\$445,000 + (-\$59,175.75) - \$5,420.35 = -\$509,596.10$$
Step 3: Calculate EAC.
$$\text{5-year Annuity Factor at 10\%} = \frac{1 - (1.10)^{-5}}{0.10} = 3.7908$$
$$\text{EAC} = \frac{-\$509,596.10}{3.7908} = \textbf{-\$134,428.18}$$
Conclusion:
The Techron II has a lower (less negative) EAC than the Techron I ($-\$134,428.18 < -\$139,178.50$). This means that on an equivalent annual basis, the Techron II is the cheaper option. You should prefer the Techron II.
Problem 20: Calculating a Bid Price
Martin Enterprises needs someone to supply it with 110,000 cartons of machine screws per year to support its manufacturing needs over the next five years, and you've decided to bid on the contract. It will cost you $940,000 to install the equipment necessary to start production; you'll depreciate this cost straight-line to zero over the project's life. You estimate that, in five years, this equipment can be salvaged for $75,000. Your fixed production costs will be $850,000 per year, and your variable production costs should be $21.43 per carton. You also need an initial investment in net working capital of $90,000. If your tax rate is 21 percent and you require a return of 12 percent on your investment, what bid price should you submit?
Solution
This is a reverse capital budgeting problem. We must find the price per unit that results in an NPV of exactly zero (Section 10.6).
Step 1: Calculate the non-OCF cash flows.
$$\text{Initial Investment (Year 0)} = -\$940,000 (\text{CapEx}) - \$90,000 (\text{NWC}) = -\$1,030,000$$
$$\text{Annual Depreciation} = \frac{\$940,000}{5} = \$188,000$$
$$\text{After-Tax Salvage (Year 5)} = \$75,000 - (\$75,000 - 0) \times 0.21 = \textbf{\$59,250}$$
$$\text{Terminal Cash Flow (Year 5)} = \$90,000 (\text{NWC Recovery}) + \$59,250 = \textbf{\$149,250}$$
Step 2: Find the required annual OCF for NPV to be zero.
We solve for the annual payment of an annuity that has a present value equal to the absolute value of the non-OCF cash flows.
$$\text{NPV} = 0 = -\$1,030,000 + \frac{OCF}{(1.12)^1} + \dots + \frac{OCF}{(1.12)^5} + \frac{\$149,250}{(1.12)^5}$$
$$\text{PV of OCF} = \$1,030,000 - \frac{\$149,250}{(1.12)^5} = \$1,030,000 - \$84,697.58 = \$945,302.42$$
$$\text{5-year Annuity Factor at 12\%} = \frac{1 - (1.12)^{-5}}{0.12} = 3.6048$$
$$\text{Required OCF} = \frac{\$945,302.42}{3.6048} = \textbf{\$262,234.33}$$
Step 3: Solve for the bid price per carton.
We use the tax shield approach and solve for sales (S).
$$\text{OCF} = (\text{Sales} - \text{Costs}) \times (1 - T_C) + \text{Depreciation} \times T_C$$
$$\$262,234.33 = (\text{S} - (\$850,000 + 110,000 \times \$21.43)) \times 0.79 + (\$188,000 \times 0.21)$$
$$\$262,234.33 = (\text{S} - \$3,207,300) \times 0.79 + \$39,480$$
$$\$222,754.33 = (\text{S} - \$3,207,300) \times 0.79$$
$$\frac{\$222,754.33}{0.79} = \text{S} - \$3,207,300$$
$$\$281,967.51 = \text{S} - \$3,207,300 \implies \text{S} = \$3,489,267.51$$
$$\text{Bid Price per Carton} = \frac{\$3,489,267.51}{110,000} = \textbf{\$31.72}$$