📖 NPV vs IRR 比较

Corporate Finance

I. Lesson Focus

This lesson examines the strengths and weaknesses of Net Present Value (NPV) and Internal Rate of Return (IRR) as capital budgeting decision tools. Candidates must understand when the two metrics give conflicting recommendations, why the conflicts occur, and why NPV is the superior criterion in mutually exclusive projects, capital rationing, and non-conventional cash-flow situations. The material emphasizes reinvestment rate assumptions, crossover rates, incremental IRR, and the mathematical conditions that produce multiple IRRs.

II. The Problem

A company is evaluating two mutually exclusive projects. Project A requires an initial outlay of CNY 1 million and generates CNY 350,000 annually for five years. Project B requires CNY 1.5 million and generates CNY 480,000 annually for five years. At a 10% required return, Project A has the higher NPV, yet its IRR is lower than Project B’s. Which project should be chosen? When capital is limited or cash-flow timing differs, do NPV and IRR always agree? This lesson resolves these conflicts and establishes clear decision rules for the CFA exam and real-world application.

III. Definitions and Calculation Mechanics

Net Present Value (NPV) measures the dollar increase in shareholder wealth. It discounts all future cash flows at the required rate of return (r) and subtracts the initial investment:

$$ NPV = \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t} - CF_0 $$

Internal Rate of Return (IRR) is the discount rate that sets NPV exactly to zero:

$$ 0 = \sum_{t=1}^{n} \frac{CF_t}{(1+IRR)^t} - CF_0 $$

IRR is typically solved by trial-and-error, financial calculator, or spreadsheet functions. Decision rule for independent projects: accept if IRR > r.

IV. Decision Consistency for Independent Projects

For independent (non-mutually exclusive) projects with conventional cash flows, NPV and IRR give identical accept/reject decisions: - NPV > 0 ⇔ IRR > r - NPV < 0 ⇔ IRR < r

Although both methods work, CFA curriculum and finance theory prefer NPV because it directly quantifies value creation in currency units.

V. Conflicts in Mutually Exclusive Projects

Conflicts arise when only one of several projects can be chosen. The two primary causes are: 1. Scale (size) differences — projects have different initial investment amounts. 2. Timing differences — the pattern of cash inflows occurs at different times.

IRR is a relative (percentage) measure; NPV is an absolute (currency) measure. When scale or timing differs, the project with the higher percentage return may not create the largest absolute wealth.

The crossover rate is the discount rate at which the NPVs of two projects are equal. To the left of the crossover rate on an NPV profile, the NPV ranking reverses the IRR ranking; to the right, the rankings agree.

VI. Incremental IRR Approach

A practical way to resolve conflicts is to compute the incremental cash flows (larger project minus smaller project) and calculate the IRR on that incremental stream.

Decision rule: if Incremental IRR > r, select the larger project; otherwise select the smaller one.
However, the most reliable method remains ranking projects by NPV and choosing the highest NPV.

VII. Why NPV Is Theoretically Superior

1. Reinvestment rate assumption: NPV assumes intermediate cash flows are reinvested at the required rate r (realistic). IRR assumes reinvestment at the project’s own IRR (often unrealistically high).
2. Value additivity: NPVs can be summed across projects; IRRs cannot.
3. Capital rationing: When funds are limited, NPV maximises total value created; IRR does not.
4. Multiple IRR problem: Non-conventional cash flows (more than one sign change) can produce multiple IRRs, rendering IRR unusable. NPV remains unambiguous.

Worked Cases

Case 1: Scale Difference Conflict

Project A: Initial outlay $500,000; $220,000 per year for 3 years.
Project B: Initial outlay $1,000,000; $420,000 per year for 3 years.
Required return r = 10%.

Calculations
NPV_A ≈ $4,860; IRR_A ≈ 15.0%
NPV_B ≈ $8,920; IRR_B ≈ 13.8%

Conclusion: Although IRR_A > IRR_B, NPV_B is larger, so choose B. The incremental IRR (B–A) ≈ 12.7% > 10%, also supports selecting the larger project.

Case 2: Cash-flow Timing Difference

Project X: –$2,000,000; +$2,800,000 (Year 1); +$100,000 (Years 2–3).
Project Y: –$2,000,000; +$200,000 (Year 1); +$1,500,000 (Years 2–3).
r = 8%.

Results
NPV_X ≈ $68,200; IRR_X ≈ 22.4%
NPV_Y ≈ $71,300; IRR_Y ≈ 19.8%

Conclusion: Choose Y on NPV. At discount rates above the crossover rate (≈14.2%), X would be preferred.

Case 3: Non-conventional Cash Flows and Multiple IRRs

Project Z (in $000): –100 (t=0), +300 (t=1), –220 (t=2).
Descartes’ rule of signs indicates up to two positive real roots. Solving yields IRRs of approximately 10% and 120%. At r = 8%, NPV_Z ≈ +$185,000 > 0. IRR is ambiguous; NPV must be used.

Traps

Key Formulas

• $NPV = \sum_{t=1}^{n} \frac{CF_t}{(1+r)^t} - CF_0$
• IRR solves $NPV(IRR) = 0$
• Crossover rate = discount rate where NPV_A = NPV_B
• Incremental IRR = IRR of (Project Larger – Project Smaller)
• Decision priority for mutually exclusive projects: NPV > IRR
• Non-conventional cash flows → possible multiple IRRs; always use NPV

Practice Questions

Q1. NPV and IRR are most likely to conflict when:
A. evaluating two independent projects with conventional cash flows
B. comparing two mutually exclusive projects with significantly different initial investment sizes
C. a single project has conventional annuity cash flows
D. the discount rate equals the IRR

Q2. Project A has positive NPV. Project B has an IRR of 12% while the required return is 10%. Both projects are mutually exclusive and of similar size. The analyst should:
A. choose A
B. choose B
C. cannot determine from the information given
D. reject both

Q3. The most serious theoretical limitation of IRR is that it:
A. cannot handle non-conventional cash flows
B. assumes intermediate cash flows are reinvested at the IRR
C. does not provide an absolute measure of value
D. is computationally difficult

Q4. A project’s cash flows change sign three times. According to Descartes’ rule of signs, the maximum number of positive real IRRs is:
A. one
B. two
C. three
D. four

Q5. When capital is rationed, a firm should select the combination of projects that maximises:
A. total IRR
B. total NPV
C. total payback
D. average profitability index

Q6. The crossover rate is best described as the discount rate at which:
A. the two projects’ IRRs are equal
B. the two projects have identical NPVs
C. each project’s IRR equals WACC
D. incremental IRR equals zero

Q7. The primary purpose of calculating an incremental IRR is to:
A. evaluate independent projects
B. resolve conflicts caused by scale or timing differences in mutually exclusive projects
C. replace NPV as the final decision criterion
D. estimate the project’s reinvestment rate

Q8. Which statement about NPV and IRR is most accurate?
A. NPV assumes reinvestment at the IRR
B. For independent projects with conventional cash flows, NPV and IRR always give the same accept/reject decision
C. NPVs cannot be added; IRRs can be added
D. IRR is always superior to NPV

Answers

Takeaways

• NPV is an absolute value measure; IRR is a relative percentage. When projects are mutually exclusive, NPV is the preferred criterion.
• Primary sources of conflict are differences in project scale and cash-flow timing; the NPV profile and crossover rate visualise the conflict.
• IRR implicitly assumes reinvestment at the project’s own IRR, which is often unrealistic; NPV’s reinvestment at the cost of capital is more reasonable.
• Non-conventional cash flows can produce multiple IRRs, making IRR unreliable—always rely on NPV in such cases.
• Incremental IRR helps diagnose conflicts but does not replace direct NPV comparison.
• Under capital rationing, the goal is to maximise aggregate NPV, not to pick projects with the highest individual IRRs.