📖 资本预算周测（10 题）

Corporate Finance

I. Lesson Focus

II. The Problem

A manufacturing firm must choose between two mutually exclusive projects: purchasing new equipment to expand capacity or upgrading the existing production line to reduce costs. The projects differ in initial outlay, project life, and risk profile, and the firm faces capital rationing. Management needs to determine which project truly increases shareholder wealth. Is it reliable to rely solely on accounting profit or simple payback? How should conflicts between NPV and IRR rankings be resolved? This lesson uses ten high-quality weekly quiz questions to systematically test and reinforce understanding of the entire capital budgeting process, enabling candidates to select the optimal answer quickly and accurately on the exam.

III. Capital Budgeting Framework and Decision Rule

Capital budgeting is one of the most critical long-term decisions in corporate finance. Its primary objective is to select projects that maximize shareholder wealth (i.e., increase firm value). The fundamental decision rule is: accept all independent projects with NPV > 0; when projects are mutually exclusive, choose the one with the highest NPV.

The capital budgeting process consists of six steps: 1. Generate investment ideas; 2. Forecast incremental cash flows; 3. Evaluate project feasibility using NPV, IRR, and other metrics; 4. Implement the selected project; 5. Perform post-audit; 6. Abandonment or replacement analysis.

IV. Correct Estimation of Project Cash Flows

Project analysis must use incremental after-tax cash flows. The operating cash flow (OCF) formula is:

$$ OCF = (S - C - D)(1 - t) + D $$

or equivalently:

$$ OCF = (S - C)(1 - t) + Dt $$

where: - S = incremental sales revenue - C = incremental cash operating costs (excluding depreciation) - D = incremental depreciation - t = marginal tax rate

Key adjustments include: - Initial investment outlay: equipment cost + installation + increase in net working capital − after-tax proceeds from sale of old assets - Terminal cash flow: after-tax salvage value + recovery of net working capital - Sunk costs are ignored - Opportunity costs must be included - Externalities (cannibalization or synergy) must be considered

V. Primary Capital Budgeting Evaluation Methods

1. Net Present Value (NPV)

$$ NPV = -CF_0 + \sum_{t=1}^{n} \frac{CF_t}{(1 + r)^t} $$ Accept if NPV > 0. NPV is the theoretically superior criterion because it directly measures the increase in shareholder wealth.

2. Internal Rate of Return (IRR)

The discount rate that makes NPV = 0. Accept if IRR > required return (r). IRR may conflict with NPV for mutually exclusive projects due to differences in scale or cash flow timing.

3. Payback Period

Time required for cumulative undiscounted cash flows to recover the initial investment. Ignores time value of money and cash flows beyond the payback period. Rule: accept if Payback < predetermined maximum.

4. Discounted Payback Period

Uses discounted cash flows to calculate recovery time. Overcomes the time-value weakness of regular payback but still ignores cash flows after the cutoff.

5. Profitability Index (PI)

$$ PI = \frac{\text{PV of Future Cash Flows}}{|CF_0|} $$ Accept if PI > 1. Under capital rationing, PI can help rank projects, but the ultimate goal remains maximization of total NPV.

VI. Resolving NPV–IRR Conflicts

Conflicts arise when projects differ in size or cash-flow timing. The crossover rate is the discount rate at which the two projects have equal NPV. In such cases, NPV is preferred because it assumes reinvestment at the cost of capital (realistic), whereas IRR assumes reinvestment at the IRR itself (often unrealistic).

Worked Cases

Case 1: Basic NPV and IRR Calculation

Project requires an initial investment of CNY 8 million and generates after-tax operating cash flows of CNY 2.2 million per year for 5 years. Cost of capital is 10%. Compute NPV and IRR.

Solution: $$ NPV = -8 + 2.2 \times \frac{1 - (1.1)^{-5}}{0.1} = -8 + 2.2 \times 3.7908 = -8 + 8.34 = 0.34 \text{ million} $$ NPV > 0, accept the project.

Using a financial calculator or Excel, IRR ≈ 12.4% > 10%, consistent conclusion.

Case 2: Mutually Exclusive Projects and Crossover Rate

Project A: Initial outlay CNY 10 million, annual cash flow CNY 4 million for 4 years.
Project B: Initial outlay CNY 16 million, annual cash flow CNY 5.5 million for 4 years. Cost of capital = 8%.

NPV_A ≈ CNY 3.128 million
NPV_B ≈ CNY 3.784 million → Choose B according to NPV.

IRR_A ≈ 21.9%, IRR_B ≈ 18.7% → IRR would incorrectly select A.

The crossover rate is approximately 14.5%. When the cost of capital is below 14.5%, NPV selects B; above 14.5%, NPV selects A.

Case 3: Comprehensive Cash Flow with Working Capital and Salvage

Equipment cost CNY 5 million, installation CNY 0.5 million, project life 4 years, salvage value CNY 0.8 million, incremental net working capital CNY 0.6 million. Annual incremental revenue CNY 3 million, cash costs CNY 1.2 million. Straight-line depreciation (zero book salvage for tax), tax rate 25%, cost of capital 10%.

Year 0 CF = −(5 + 0.5) − 0.6 = −6.1 million
Annual OCF = (3 − 1.2 − 1.375)(1 − 0.25) + 1.375 = 1.60625 million
Terminal Year 4 CF = 0.8 × (1 − 0.25) + 0.6 = 1.2 million
Total Year 4 CF = 1.60625 + 1.2 = 2.80625 million

Calculated NPV ≈ CNY 0.684 million > 0, so the project is acceptable.

Traps

Key Formulas

• OCF = (S − C)(1 − t) + Dt
• NPV = −CF₀ + Σ [CFₜ / (1+r)ᵗ]
• IRR: discount rate where NPV = 0
• PI = PV(future cash flows) / Initial investment
• Discounted Payback: time when cumulative discounted cash flows equal initial outlay
• Crossover Rate: discount rate at which two projects have identical NPV
• NPV equals the direct increase in shareholder wealth (primary decision criterion)

Practice Questions

Q1. A project requires an initial investment of $2 million and generates after-tax cash flows of $0.55 million per year for 6 years. The cost of capital is 9%. The project’s NPV is closest to:
A. $0.184 million　B. $0.257 million　C. $0.321 million　D. $0.416 million

Q2. Which of the following is NOT an incremental cash flow?
A. Cannibalization loss caused by the new project
B. Market research fee paid two years ago
C. Recovery of net working capital at project end
D. Opportunity cost of using company-owned land

Q3. When NPV and IRR rankings conflict for two mutually exclusive projects, the correct decision criterion is:
A. The project with the higher IRR
B. The project with the higher NPV
C. The project with the shorter payback
D. The project with the higher PI

Q4. A project has an initial outlay of $5 million and cash flows of $1.0m, $1.5m, $2.0m, $1.8m, and $1.7m over the next five years. Cost of capital is 10%. The discounted payback period is closest to:
A. 3.2 years　B. 3.6 years　C. 4.1 years　D. 4.5 years

Q5. Under capital rationing, a firm should choose the combination of projects that:
A. Maximizes the sum of IRRs
B. Maximizes the sum of PIs
C. Maximizes total NPV
D. Minimizes total payback periods

Q6. A project has an IRR of 15% and a cost of capital of 12%, yet its NPV is negative. The most likely reason is:
A. The project has unconventional cash flows (multiple sign changes)
B. The project is much smaller than the firm’s average project
C. Reinvestment rate assumption differences
D. Depreciation tax shield was omitted in the calculation

Q7. Which statement about the profitability index (PI) is correct?
A. When PI = 1, NPV = 0
B. A higher PI always means a higher NPV
C. Ranking by PI under capital rationing always yields the highest total NPV
D. PI is only useful for independent projects and never for mutually exclusive projects

Q8. A firm is considering replacing old equipment with a book value of $0.3 million. The old equipment can be sold today for $0.45 million. Tax rate is 25%. If the new equipment costs $1.8 million before considering the sale of the old asset, the initial cash outflow (in millions) to be used in the NPV analysis is closest to:
A. 1.35　B. 1.5375　C. 1.6125　D. 1.80

Answers

Takeaways

• NPV is the gold standard in capital budgeting and equals the direct increase in shareholder wealth
• Cash flows must be incremental, after-tax, and include opportunity costs and externalities
• Sunk costs are never included in decision making
• In mutually exclusive projects, always follow NPV when it conflicts with IRR
• Non-conventional cash flows can render IRR unreliable
• Under capital rationing the objective is to maximize total NPV, not any single relative metric