📖 IRR（内部收益率）法

Corporate Finance

I. Lesson Focus

II. The Problem

A company is evaluating a new factory requiring an initial outlay of CNY 8 million and generating uneven cash inflows over the next five years. Management wants to know the project’s true annualized rate of return. If the firm’s cost of capital is 10%, should the project be accepted? Traditional payback period and accounting rate of return cannot answer the question of “what is the true annualized yield?” The IRR was developed precisely to solve the core problem: “What discount rate makes the project’s net present value equal to zero?” In the CFA exam, IRR is one of the most frequently tested capital-budgeting tools, yet it contains numerous traps, especially multiple IRRs arising from non-conventional cash flows.

III. Basic Concept and Calculation Principle of IRR

The Internal Rate of Return (IRR) is the discount rate that makes the present value of a project’s future cash flows equal to the present value of its initial investment (i.e., NPV = 0). Economically, it represents the project’s expected compound annualized rate of return.

Mathematically, solve for $r$ in: $$ NPV = CF_0 + \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \cdots + \frac{CF_n}{(1+r)^n} = 0 $$ where $CF_0$ is typically the negative initial outlay.

IRR Decision Rule
- Accept the project if IRR > cost of capital (WACC or required return).
- Reject if IRR < cost of capital.
- Indifferent if IRR = cost of capital.

Compared with NPV, IRR’s advantage is that it expresses return as an intuitive percentage that can be compared directly with financing cost. Its main disadvantages are the unrealistic reinvestment-rate assumption (reinvestment at the IRR itself) and potential mathematical problems with non-conventional cash flows.

IV. Methods of Calculating IRR

For conventional cash flows (initial outflow followed by a series of inflows), IRR is usually found by trial-and-error, financial calculator, or Excel’s IRR function. CFA exams frequently require candidates to perform linear interpolation manually.

Interpolation formula: $$ IRR \approx r_L + \left( \frac{NPV_L}{NPV_L - NPV_H} \right) \times (r_H - r_L) $$ where $r_L$ is the lower trial rate that produces positive NPV and $r_H$ is the higher trial rate that produces negative NPV.

V. Non-conventional Cash Flows and the Multiple-IRR Problem

When the sign of cash flows changes more than once during the project’s life, multiple IRRs may exist because an $n$th-degree polynomial can have up to $n$ real roots.

Descartes’ Rule of Signs: The number of sign changes equals the maximum number of positive real roots.

Solutions: 1. Use Modified IRR (MIRR): compound all inflows to a terminal value at the cost of capital, discount all outflows to present value at the financing cost, then solve for the single rate. 2. Rely on NPV as the primary decision criterion.

No-IRR situation: When all cash flows are positive or all negative, the equation has no real roots.

VI. Relationship Between IRR and NPV (Complementarity and Conflict)

• Conventional independent projects: IRR and NPV give the same accept/reject decision.
• Mutually exclusive projects: When project sizes or cash-flow timing differ, a crossover rate may exist and rankings can conflict. In such cases, NPV is the superior criterion.
• NPV Profile: Plot NPV on the vertical axis against discount rate on the horizontal axis. The horizontal intercept is the IRR.

Worked Cases

Case 1: IRR Calculation for a Conventional Project

Project: Initial investment CNY 1,000,000; cash inflows CNY 400,000, 500,000, and 600,000 over the next three years. Cost of capital = 12%. Compute IRR and decide.

Trial-and-error: - At 18%: NPV ≈ –100 + 33.90 + 35.92 + 36.55 ≈ +6.37 (>0) - At 22%: NPV ≈ –100 + 32.79 + 33.58 + 33.00 ≈ –0.63 (<0)

Interpolation: $$ IRR \approx 18\% + \left( \frac{6.37}{6.37+0.63} \right) \times 4\% \approx 21.64\% $$ Since 21.64% > 12%, accept the project.

Case 2: Multiple IRRs with Non-conventional Cash Flows

Cash flows (CNY 000): Year 0: –100; Year 1: +300; Year 2: –220.

Sign changes twice → up to two positive real roots. Solving the quadratic yields IRR₁ = 10% and IRR₂ = 100%. IRR is meaningless here. At a 10% cost of capital, NPV ≈ +49.59 > 0, so the project should be accepted on NPV grounds.

Case 3: IRR–NPV Conflict in Mutually Exclusive Projects

Project A: Investment CNY 2 m, IRR = 18%, NPV(at 10%) = CNY 0.52 m
Project B: Investment CNY 8 m, IRR = 15%, NPV(at 10%) = CNY 1.48 m

Although A has the higher IRR, B has the materially higher NPV. If the projects are mutually exclusive, select B. The crossover rate is approximately 13.8%. When the cost of capital is below 13.8%, NPV prefers B; above 13.8%, IRR prefers A.

Traps

Key Formulas

• IRR definition: discount rate where $NPV(r) = 0$
• Linear interpolation: $IRR \approx r_L + \frac{NPV_L}{NPV_L-NPV_H}(r_H-r_L)$
• MIRR: $\text{MIRR} = \sqrt[n]{\frac{FV(\text{inflows at WACC})}{PV(\text{outflows at financing cost})}} - 1$
• Decision rule for independent projects: accept if $IRR > r_c$
• When IRR and NPV conflict, NPV is preferred

Practice Questions

Q1. Which of the following is the most accurate definition of IRR?
A. The project’s expected accounting rate of return
B. The discount rate that makes the project’s NPV equal to zero
C. The ratio of the present value of cash flows to initial investment
D. The growth rate at which the project breaks even

Q2. A conventional project has an IRR of 15% and the company’s WACC is 12%. The project should:
A. Be rejected
B. Have a negative NPV
C. Be accepted
D. Have an MIRR necessarily lower than 15%

Q3. When cash-flow signs change twice during a project’s life, the most likely outcome is:
A. A single IRR
B. Multiple IRRs
C. No IRR
D. A negative IRR

Q4. For two mutually exclusive projects, when the cost of capital is below the crossover rate, it is usually true that:
A. The higher-IRR project also has the higher NPV
B. The higher-NPV project has the lower IRR
C. Both criteria give the same ranking
D. The IRR rule should be followed

Q5. A project has cash flows of –200, +800, –650. Which statement is most accurate?
A. The project cannot have an IRR
B. The project can have at most two IRRs
C. The project’s IRR must be greater than zero
D. The project should be accepted if its IRR exceeds the cost of capital

Q6. Which of the following is not a limitation of the IRR rule?
A. Unrealistic reinvestment-rate assumption
B. Potential for multiple solutions with non-conventional cash flows
C. Inability to rank projects of different sizes
D. Always consistent with the NPV decision

Q7. The most accurate way to calculate IRR is to:
A. Interpolate using only 10% and 20% test rates
B. Use a financial calculator or Excel IRR function after verifying cash-flow signs
C. Divide average annual cash flow by initial investment
D. Rely solely on the payback period

Q8. A project requires an initial outlay of 100 and then 30 per year for five years. NPV at 10% ≈ 19.0, at 15% ≈ 3.5, at 18% ≈ –4.2. The IRR is closest to:
A. 16.2%
B. 16.8%
C. 17.1%
D. 15.9%

Answers

Takeaways

• IRR is the discount rate that sets NPV to zero and represents the project’s expected compound annual return
• For conventional independent projects, accept when IRR exceeds the cost of capital
• Non-conventional cash flows can produce multiple IRRs; NPV is then the reliable criterion
• In mutually exclusive projects, IRR and NPV may rank differently; always prefer NPV
• IRR implicitly assumes reinvestment at the IRR itself, which is its main theoretical weakness
• Always draw the cash-flow timeline first to determine whether the project is conventional or non-conventional