📖 资本预算：互斥项目 vs 独立项目

Corporate Finance

I. Lesson Focus

This lesson explains the critical distinction between mutually exclusive and independent capital projects, the appropriate decision rules under both unlimited and capital-constrained budgets, the reasons for NPV-IRR ranking conflicts, and the correct application of the profitability index. Mastery of these concepts is essential for both corporate finance valuation and CFA exam questions involving project selection.

II. The Problem

A manufacturing company receives two investment proposals at the same time. Project A is a new automated production line and Project B is a warehouse expansion. Both require substantial capital, but the firm’s annual capital budget is limited, so only one can be chosen. At the same time, the company has several smaller equipment-upgrade projects that can be implemented simultaneously as long as each meets the minimum return requirement. How should management distinguish “choose only one” mutually exclusive projects from “can do all” independent projects? When capital is limited, which metric should be used to rank them? Choosing the wrong rule can destroy shareholder value. This lesson systematically solves these practical and exam-critical issues.

III. Definitions and Distinction Between Mutually Exclusive and Independent Projects

Mutually exclusive projects are investment opportunities in which acceptance of one project precludes acceptance of the others, usually because of resource, technical, or strategic constraints. The defining feature is “choose only one.”

Independent projects are opportunities in which the acceptance or rejection of one project does not affect the decision on any other project. Multiple independent projects can be accepted together provided each individually satisfies the decision criterion.

Core distinction: - Mutually exclusive projects have direct conflict (resource or functional substitution). - Independent projects have no conflict and may coexist.

Project classification in capital budgeting directly determines the decision process and ranking method used.

IV. Decision Rules Without Capital Constraints

1. Independent Projects

• Decision rule: Accept any project with NPV > 0 or IRR > required rate of return (r).
• All qualifying independent projects can be accepted simultaneously.
• NPV is preferred because it directly measures the absolute increase in shareholder wealth (in currency units).

2. Mutually Exclusive Projects

• Only one project can be chosen.
• Correct rule: Select the project with the highest NPV.
• Even if one project has a higher IRR, it is rejected if its NPV is lower (due to differences in scale or cash-flow timing).

Key principle: NPV is an absolute value-creation metric; IRR is a relative rate-of-return metric. When projects differ in size, life, or cash-flow pattern, IRR can mislead.

V. Decision Making Under Capital Rationing

When available capital is less than the total required by all positive-NPV projects, ranking becomes necessary.

• Independent projects + capital rationing: Rank by Profitability Index (PI). $$ PI = \frac{\text{PV of Future Cash Flows}}{\text{Initial Investment}} = 1 + \frac{\text{NPV}}{\text{Initial Investment}} $$ Select projects in descending order of PI until capital is exhausted.

• Mutually exclusive projects + capital rationing: NPV remains the primary criterion; consider incremental effects. PI should not be used to rank mutually exclusive projects because they cannot be implemented together.

VI. Sources of Decision Conflicts (NPV vs IRR Ranking Conflicts) and Resolution

When NPV and IRR rank mutually exclusive projects differently, the conflict usually arises from:

1. Scale (size) discrepancy: A larger project may have higher NPV while a smaller project has higher IRR.
2. Cash-flow timing discrepancy: Projects with cash flows received earlier tend to have higher IRR; projects with later cash flows may have higher NPV at lower discount rates.

Resolution methods: - Always use NPV as the final decision criterion (value maximization). - Draw the NPV Profile for both projects. The intersection is the Fisher intersection. To the left of the intersection, IRR ranking may be valid; to the right, NPV is more reliable. - Use Incremental IRR: Compute the IRR on the difference in cash flows between the two projects. If incremental IRR > cost of capital, choose the larger project.

Worked Cases

Case 1: Mutually Exclusive Projects Without Capital Constraints (Scale Difference)

Project A: Initial outlay CNY 8 million, annual cash flow CNY 2.5 million for 5 years.
Project B: Initial outlay CNY 15 million, annual cash flow CNY 4.2 million for 5 years.
Cost of capital r = 10%.

Calculations: - Project A: NPV = –8 + 2.5 × 3.7908 ≈ +1.477 million; IRR ≈ 16.8%. - Project B: NPV = –15 + 4.2 × 3.7908 ≈ +0.921 million; IRR ≈ 12.9%.

Decision: Although Project A has the higher IRR, the correct choice is the project with the higher NPV. In this example Project A is selected (NPV 1.477 > 0.921). IRR misleads when scale differs.

Case 2: Cash-Flow Timing Difference Causing Conflict

Project X: Initial investment $1 million, cash flow $1.3 million in Year 1 (front-loaded).
Project Y: Initial investment $1 million, roughly equal annual cash flows of approximately $0.45 million for 3 years (back-loaded).

At r = 10%: - NPV_X ≈ $0.1818 million, IRR_X = 30%. - NPV_Y ≈ $0.1157 million, IRR_Y ≈ 15.8% (illustrative).

At a lower discount rate of 5%, Project Y’s NPV may exceed Project X’s.
Decision: At the firm’s 10% cost of capital, select the higher-NPV project (X). If the cost of capital falls below the Fisher intersection, Y could become preferable. On the exam, always use NPV at the given discount rate.

Case 3: Capital Rationing with Independent Projects (PI Application)

Available capital = $10 million. Independent projects:

Optimal combination: Accept A and B (total investment $7m, total NPV $3.0m). The remaining $3m is insufficient for C. Projects C and D are rejected.
Ranking by IRR alone might incorrectly favor C, but PI ranking maximizes value created per dollar of scarce capital.

Traps

Key Formulas

• $NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$
• $PI = 1 + \frac{NPV}{\text{Initial Investment}}$
• Mutually exclusive decision: Select project with $\max(NPV)$
• Independent project decision: Accept all projects where $NPV>0$ or $IRR>r$
• Under capital rationing for independent projects: Rank by descending $PI$ until capital is exhausted
• Incremental IRR: IRR calculated on the differential cash flows of two mutually exclusive projects; accept larger project if incremental IRR > r

Practice Questions

Q1. Which of the following is most likely a mutually exclusive project situation?
A. Purchasing two different brands of printers
B. Building either a warehouse or an office building on the same plot of land
C. Upgrading ERP systems at two unrelated factories
D. Buying raw materials and office supplies

Q2. In the absence of capital constraints, the most appropriate decision rule for independent projects is to accept:
A. All projects with IRR > 0
B. All projects with NPV > 0
C. All projects with PI > 1.0 but only the highest-PI project
D. Projects ranked by highest IRR until capital is exhausted

Q3. Two mutually exclusive projects A and B have NPV_A = $1.2m, NPV_B = $1.5m, IRR_A = 18%, IRR_B = 14%, and cost of capital = 10%. The company should choose:
A. Project A
B. Project B
C. Both projects
D. Neither project

Q4. A company faces capital rationing with only $6 million available. For the following independent projects, the optimal selection method is:
A. Select the single project with the highest NPV
B. Rank by IRR
C. Rank by PI until funds are exhausted
D. Accept all projects with NPV > 0

Q5. The main reasons NPV and IRR can give conflicting rankings for mutually exclusive projects are:
A. Different accounting rates of return
B. Differences in project scale or cash-flow timing
C. Different tax rates
D. Different depreciation methods

Q6. The profitability index (PI) is most useful when:
A. Choosing among mutually exclusive projects with no capital constraint
B. Selecting independent projects under capital rationing
C. Making any mutually exclusive project decision
D. Comparing projects with different economic lives

Q7. Which statement about incremental IRR is correct?
A. It is always the weighted average of the two projects’ IRRs
B. If incremental IRR > cost of capital, the larger project should be accepted
C. It is used only when NPVs are equal
D. It is used to rank independent projects

Q8. Two mutually exclusive projects have opposite NPV and IRR rankings at the current cost of capital. If the cost of capital decreases, the decision on the side of the NPV-profile crossover point should primarily be based on:
A. IRR
B. PI
C. NPV
D. Payback period

Answers

Takeaways

• Mutually exclusive projects require selection of the single highest-NPV alternative.
• Independent projects can all be accepted provided each has NPV > 0.
• Under capital rationing, rank independent projects by PI; mutually exclusive projects are still ranked by NPV.
• When NPV and IRR conflict, trust NPV.
• The Fisher intersection on the NPV profile indicates the discount rate at which rankings switch; lower discount rates favor back-loaded cash flows.
• Incremental IRR is a practical tool for resolving scale conflicts.