公司金融（Corporate Finance）
一、本课定位
| 课次 | 主题 | 能力 |
|---|---|---|
| L276 | 资本预算综合练习 | 综合运用NPV、IRR、Payback、PI、EAA等方法进行项目决策，能识别互斥项目、资本限额、项目风险调整等情景下的最优选择 |
二、我们要解决什么问题？
某制造企业同时收到三个独立资本投资提案：A项目初始投资800万元，预计每年产生不等额现金流，项目期5年；B项目初始投资1200万元，现金流稳定但IRR略低于公司WACC；C项目与A项目互斥，规模更大但NPV在高折现率下更优。公司资本预算有限，且管理层同时关心回收速度和会计回报率。如何在NPV、IRR、回收期、盈利指数等多准则下做出一致且正确的资本配置决策？这就是本课要综合训练的核心能力。
三、资本预算核心决策准则回顾
资本预算的核心目标是最大化股东财富，最优决策准则是净现值（NPV）。NPV>0的项目接受，NPV<0的项目拒绝。
当面对互斥项目时，必须选择NPV更高的那个，而非IRR更高的那个，这是CFA最常考的陷阱。
主要决策指标公式： - NPV = $\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$ - IRR：使NPV=0的折现率，求解$\sum_{t=0}^{n} \frac{CF_t}{(1+IRR)^t}=0$ - 回收期（Payback Period）：累计现金流达到初始投资所需年限（非折现） - 折现回收期（Discounted Payback）：累计折现现金流达到初始投资所需年限 - 盈利指数（PI）= $\frac{\text{PV of future CFs}}{\text{Initial Investment}}$，PI>1接受 - 会计收益率（AAR）= $\frac{\text{Average Net Income}}{\text{Average Book Value}}$
四、互斥项目与规模差异问题
当两个项目初始投资规模不同或现金流发生时间不同时，NPV与IRR可能给出冲突结论。此时必须以NPV为准。
解决方法包括：
1. 增量IRR（Incremental IRR）：计算两个项目现金流差额的IRR，若增量IRR > WACC，则选择较大项目。
2. 等值年金法（EAA, Equivalent Annual Annuity）：将NPV转化为每年等额年金，便于比较不同寿命项目。
五、资本限额（Capital Rationing）下的决策
当可用资本少于所有正NPV项目所需资金总和时，不能简单选NPV最大的项目，而应选择单位资本NPV最高（或PI最高）的项目组合，使有限资本创造最大总NPV。
六、项目风险调整
高风险项目应使用更高的折现率（调整后WACC或纯权益成本）。CFA常通过提高折现率考察考生是否仍能正确判断NPV符号。
完整案例演算
案例 1：NPV vs IRR冲突（互斥项目）
公司WACC=10%。
项目A：初始投资$500,000，年现金流第1-4年均为$180,000。
项目B：初始投资$800,000，年现金流第1-4年均为$280,000。
计算过程：
NPV_A = -500,000 + 180,000×(1.1⁻¹ + 1.1⁻² + 1.1⁻³ + 1.1⁻⁴) = -500,000 + 180,000×3.1699 ≈ $70,582
NPV_B = -800,000 + 280,000×3.1699 ≈ $87,572
IRR_A ≈ 16.0%，IRR_B ≈ 14.8%（通过财务计算器或Excel IRR函数求得）。
决策：虽然IRR_A > IRR_B，但NPV_B > NPV_A，应选择项目B。冲突原因在于项目B规模更大。
案例 2：资本限额下的盈利指数排序
可用资本仅1,500万元。四个独立项目数据如下：
| 项目 | 初始投资(万元) | PV of future CFs(万元) | NPV(万元) | PI |
|---|---|---|---|---|
| A | 400 | 520 | 120 | 1.30 |
| B | 600 | 810 | 210 | 1.35 |
| C | 700 | 840 | 140 | 1.20 |
| D | 500 | 650 | 150 | 1.30 |
最优组合：先按PI排序：B(1.35)、A(1.30)、D(1.30)、C(1.20)。
选择B+A+D=1,500万元，总NPV=210+120+150=480万元。
若选B+C+D=1,800万元超支，故放弃。PI法在此情景下优于单纯NPV排序。
案例 3：不同寿命项目的EAA比较
项目X：投资100万元，寿命3年，每年CF 45万元，WACC=8%，NPV≈18.82万元。
项目Y：投资160万元，寿命5年，每年CF 48万元，NPV≈32.47万元。
EAA_X = NPV_X / PV annuity factor(3年,8%) = 18.82 / 2.5771 ≈ 7.30万元
EAA_Y = 32.47 / 3.9927 ≈ 8.13万元
决策：虽然NPV_Y更高，但EAA_Y > EAA_X，重复投资时Y每年创造更多价值，应选择Y。
易错陷阱对照
| 陷阱场景 | 错误做法 | 正确做法 | CFA常考点 |
|---|---|---|---|
| 互斥项目IRR与NPV冲突 | 选择IRR更高的项目 | 始终选择NPV更高的项目 | 最核心陷阱 |
| 资本限额情景 | 单纯挑选NPV最大的项目 | 按PI从高到低排序选择组合 | 易忽略组合优化 |
| 非常规现金流（多重符号变化） | 直接报告单一IRR | 可能存在多重IRR，优先看NPV | Descartes符号法则 |
| 不同项目寿命 | 直接比NPV | 使用EAA或最小公倍数法 | 寿命不一致未调整 |
| 仅计算回收期 | 认为回收期短的项目更好 | 回收期仅作辅助，NPV优先 | 忽略货币时间价值 |
| 使用AAR决策 | 把AAR与WACC比较 | AAR不是现金流指标，不能与WACC比 | 会计收益率陷阱 |
关键公式 / 关系速记
- NPV = $\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$
- PI = 1 + (NPV / Initial Investment)
- Incremental IRR：对ΔCF序列求IRR
- EAA = $\frac{r \times NPV}{1-(1+r)^{-n}}$
- Discounted Payback：累计PV(CF)首次≥初始投资的时间
- 当现金流符号变化次数>1时，可能存在多个IRR（Descartes法则）
- 资本限额下：最大化总NPV/单位资本（即PI排序）
练习题（含计算与情景）
Q1. 某互斥项目A的NPV为$120,000，IRR为14%；项目B的NPV为$150,000，IRR为12%，WACC=10%。应选择哪个项目？
A. 项目A
B. 项目B
C. 两者均可
D. 无法判断
Q2. 在资本限额下，最优排序指标是：
A. NPV
B. IRR
C. 盈利指数(PI)
D. 回收期
Q3. 以下哪种情况最可能导致NPV与IRR决策冲突？
A. 常规现金流且规模相同
B. 互斥项目且初始投资规模显著不同
C. 两个项目寿命均为5年
D. 折现率等于IRR
Q4. 项目现金流符号变化3次，根据Descartes符号法则，最多可能有几个正实数IRR？
A. 1个
B. 2个
C. 3个
D. 4个
Q5. 某项目初始投资200万元，第1-5年现金流分别为80、70、60、50、40万元，WACC=10%。其PI最接近：
A. 0.85
B. 1.12
C. 1.25
D. 1.38
Q6. 使用等值年金法（EAA）的主要目的是：
A. 比较不同初始投资规模的项目
B. 比较不同经济寿命的项目
C. 计算回收期
D. 调整项目风险
Q7. 若公司采用折现回收期决策，将导致：
A. 接受过多高风险长期项目
B. 偏好前期现金流多的项目
C. 与NPV决策完全一致
D. 忽略货币时间价值
Q8. 在资本预算中，会计收益率（AAR）的主要缺陷是：
A. 没有考虑货币的时间价值
B. 计算过于复杂
C. 无法与WACC直接比较
D. A和C都对
答案与详解
| 题号 | 答案 | 详解 |
|---|---|---|
| Q1 | B | NPV是首要决策准则，B的NPV更高，应选B。尽管A的IRR更高，但互斥项目必须服从NPV。 |
| Q2 | C | 资本限额下，PI（单位资本创造的现值）是最佳排序指标，可使有限资金产生最大总NPV。 |
| Q3 | B | 规模或现金流发生时间差异大的互斥项目最易出现NPV-IRR冲突。 |
| Q4 | C | 符号变化次数为3，最多有3个正实根IRR。 |
| Q5 | B | PV of inflows ≈ 80/1.1 + 70/1.1² + ... ≈ 224.6万元，PI = 224.6/200 ≈ 1.123，最接近1.12。 |
| Q6 | B | EAA专门用于将不同寿命项目的NPV转化为可比的年度等值，便于比较。 |
| Q7 | B | 折现回收期仍优先考虑早期现金流，因此会偏好前期回本快的项目。 |
| Q8 | D | AAR基于会计利润和账面价值，既不考虑时间价值，也不能与WACC（现金流折现率）直接比较。 |
本节要点速记
- NPV永远是资本预算第一决策准则，互斥项目冲突时服从NPV。
- 资本限额情景下必须使用盈利指数（PI）排序选择最优项目组合。
- 不同寿命项目需用EAA或最小公倍数法进行比较。
- 非常规现金流可能产生多重IRR，此时应优先依赖NPV。
- 回收期和AAR仅作辅助指标，不能替代NPV。
- 风险较高的项目应使用更高的折现率，NPV仍为最终判断依据。
Corporate Finance
I. Lesson Focus
| Lesson | Topic | Capability |
|---|---|---|
| L276 | Capital Budgeting Practice | Integrate NPV, IRR, Payback, PI, EAA and other tools to make project decisions; identify optimal choices under mutually exclusive projects, capital rationing, and risk-adjusted discount rates |
II. The Problem
A manufacturing company receives three independent capital investment proposals. Project A requires an initial outlay of CNY 8 million and generates uneven cash flows over 5 years. Project B requires CNY 12 million with stable cash flows, yet its IRR is slightly below the company’s WACC. Project C is mutually exclusive with A, has a larger scale, and shows a higher NPV at elevated discount rates. The firm faces a limited capital budget, and management also cares about payback speed and accounting returns. How should the firm reach a consistent and value-maximizing decision when multiple criteria (NPV, IRR, payback, profitability index) conflict? This lesson provides comprehensive practice on exactly these integrated capital-budgeting skills.
III. Review of Core Capital Budgeting Decision Rules
The primary goal of capital budgeting is to maximize shareholder wealth. The superior decision rule is Net Present Value (NPV). Accept projects with NPV > 0; reject those with NPV < 0.
When ranking mutually exclusive projects, always select the one with the higher NPV, not the higher IRR. This is one of the most frequently tested CFA traps.
Key decision metrics and formulas: - NPV = $\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$ - IRR solves $\sum_{t=0}^{n} \frac{CF_t}{(1+IRR)^t}=0$ - Payback Period: years required for undiscounted cumulative cash flows to recover initial investment - Discounted Payback: years required for cumulative discounted cash flows to recover initial investment - Profitability Index (PI) = $\frac{\text{PV of future CFs}}{\text{Initial Investment}}$; accept if PI > 1 - Accounting Rate of Return (AAR) = $\frac{\text{Average Net Income}}{\text{Average Book Value}}$
IV. Mutually Exclusive Projects and Scale Differences
When projects differ in size or cash-flow timing, NPV and IRR can give conflicting rankings. NPV must prevail.
Two common remedies:
1. Incremental IRR: compute IRR on the difference in cash flows between the larger and smaller project. If incremental IRR > WACC, choose the larger project.
2. Equivalent Annual Annuity (EAA): convert each project’s NPV into an annual annuity payment to allow direct comparison of projects with unequal lives.
V. Capital Rationing
When available capital is insufficient to fund all positive-NPV projects, do not simply pick the projects with the largest individual NPVs. Instead, rank by NPV per unit of capital (or PI) and select the combination that maximizes total NPV within the budget constraint.
VI. Risk Adjustment
Higher-risk projects require higher discount rates (adjusted WACC or pure-play cost of equity). CFA questions often test whether candidates still correctly sign the NPV after the discount rate is increased.
Worked Cases
Case 1: NPV vs IRR Conflict (Mutually Exclusive Projects)
WACC = 10%.
Project A: Initial investment $500,000; annual CF $180,000 for 4 years.
Project B: Initial investment $800,000; annual CF $280,000 for 4 years.
Calculations:
NPV_A = –500,000 + 180,000 × (1.1⁻¹ + 1.1⁻² + 1.1⁻³ + 1.1⁻⁴) = –500,000 + 180,000 × 3.1699 ≈ $70,582
NPV_B = –800,000 + 280,000 × 3.1699 ≈ $87,572
IRR_A ≈ 16.0%, IRR_B ≈ 14.8% (obtained via financial calculator or Excel IRR function).
Decision: Although IRR_A > IRR_B, NPV_B > NPV_A, so select Project B. The conflict arises because Project B is larger in scale.
Case 2: Capital Rationing and Profitability Index Ranking
Budget constraint = $15 million. Project data (in millions):
| Project | Initial Outlay | PV of Future CFs | NPV | PI |
|---|---|---|---|---|
| A | 4 | 5.2 | 1.2 | 1.30 |
| B | 6 | 8.1 | 2.1 | 1.35 |
| C | 7 | 8.4 | 1.4 | 1.20 |
| D | 5 | 6.5 | 1.5 | 1.30 |
Optimal combination: Rank by PI: B (1.35), A (1.30), D (1.30), C (1.20).
Select B + A + D = exactly $15 million; total NPV = 2.1 + 1.2 + 1.5 = $4.8 million.
Selecting B + C + D would exceed the budget, so it is rejected. PI ranking outperforms pure NPV ranking under capital rationing.
Case 3: EAA for Projects with Unequal Lives
Project X: Investment $1.0 million, 3-year life, annual CF $0.45 million, WACC = 8%, NPV ≈ $0.1882 million.
Project Y: Investment $1.6 million, 5-year life, annual CF $0.48 million, NPV ≈ $0.3247 million.
EAA_X = 0.1882 / PV annuity factor(3 yrs, 8%) = 0.1882 / 2.5771 ≈ $0.0730 million
EAA_Y = 0.3247 / 3.9927 ≈ $0.0813 million
Decision: Although NPV_Y is larger, EAA_Y > EAA_X, so Project Y creates more annual value when projects are repeated. Choose Y.
Traps
| Trap Scenario | Common Mistake | Correct Approach | CFA Favorite |
|---|---|---|---|
| NPV–IRR conflict on mutually exclusive projects | Pick higher-IRR project | Always choose higher-NPV project | Highest-frequency trap |
| Capital rationing | Select largest individual NPVs | Rank by PI and optimize combination within budget | Overlooked combinatorial optimization |
| Non-conventional cash flows (multiple sign changes) | Report single IRR | Multiple IRRs possible; rely on NPV | Descartes’ rule of signs |
| Unequal project lives | Compare NPVs directly | Use EAA or least common multiple of lives | Failure to adjust for different durations |
| Relying solely on payback | Prefer shortest payback | Payback is only supplementary; NPV rules | Ignoring time value of money |
| Using AAR for accept/reject | Compare AAR with WACC | AAR is an accounting measure, not comparable to WACC | Accounting-rate-of-return trap |
Key Formulas
- NPV = $\sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$
- PI = 1 + (NPV / Initial Investment)
- Incremental IRR: IRR calculated on the differential cash-flow series
- EAA = $\frac{r \times NPV}{1-(1+r)^{-n}}$
- Discounted Payback: time when cumulative PV of CFs first ≥ initial outlay
- Number of positive real IRRs ≤ number of sign changes in cash-flow series (Descartes’ rule)
- Under capital rationing: maximize total NPV per dollar of scarce capital (i.e., rank by PI)
Practice Questions
Q1. Mutually exclusive Project A has NPV = $120,000 and IRR = 14%; Project B has NPV = $150,000 and IRR = 12%. WACC = 10%. Which project should be chosen?
A. Project A
B. Project B
C. Both
D. Cannot determine
Q2. Under capital rationing, the best ranking criterion is:
A. NPV
B. IRR
C. Profitability Index (PI)
D. Payback period
Q3. Which situation is most likely to produce conflicting NPV and IRR rankings?
A. Conventional cash flows of identical size
B. Mutually exclusive projects with significantly different initial investments
C. Both projects last exactly 5 years
D. Discount rate equals IRR
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. 1
B. 2
C. 3
D. 4
Q5. A project requires a $2 million initial investment and produces cash flows of $0.8 m, $0.7 m, $0.6 m, $0.5 m, and $0.4 m over the next five years. WACC = 10%. Its PI is closest to:
A. 0.85
B. 1.12
C. 1.25
D. 1.38
Q6. The main purpose of the Equivalent Annual Annuity (EAA) method is to:
A. Compare projects with different initial investment sizes
B. Compare projects with different economic lives
C. Calculate the payback period
D. Adjust for project risk
Q7. Using the discounted payback criterion tends to:
A. Accept too many high-risk, long-term projects
B. Favor projects that generate cash flows earlier
C. Produce decisions identical to NPV
D. Ignore the time value of money
Q8. The primary weakness of the accounting rate of return (AAR) is that it:
A. Ignores the time value of money
B. Is too complicated to calculate
C. Cannot be compared directly with WACC
D. Both A and C
Answers
| Question | Answer | Explanation |
|---|---|---|
| Q1 | B | NPV is the primary decision criterion. Project B has the higher NPV and should be chosen even though Project A has a higher IRR. |
| Q2 | C | Under capital rationing, PI (present value created per dollar invested) produces the combination that maximizes total NPV within the budget. |
| Q3 | B | Large differences in scale or cash-flow timing between mutually exclusive projects are the most common source of NPV–IRR conflict. |
| Q4 | C | With three sign changes, up to three positive real IRRs are mathematically possible. |
| Q5 | B | PV of inflows ≈ $2.246 million; PI = 2.246 / 2.0 ≈ 1.123, closest to 1.12. |
| Q6 | B | EAA converts NPVs of projects with unequal lives into comparable annual amounts. |
| Q7 | B | Discounted payback still emphasizes earlier cash flows and therefore favors front-loaded projects. |
| Q8 | D | AAR is based on accounting profit and book values; it ignores time value and cannot be compared with WACC (a cash-flow discount rate). |
Takeaways
- NPV is always the dominant capital-budgeting criterion; when NPV and IRR conflict on mutually exclusive projects, follow NPV.
- Under capital rationing, rank projects by PI and select the optimal combination that stays inside the budget.
- Use EAA (or the least-common-multiple approach) when project lives differ.
- Non-conventional cash flows may produce multiple IRRs; rely on NPV in such cases.
- Payback and AAR are supplementary only and must never replace NPV.
- Increase the discount rate for riskier projects, but still make the final accept/reject decision using NPV.