公司金融（Corporate Finance）
一、本课定位
| 课次 | 主题 | 能力 |
|---|---|---|
| L272 | 折现回收期法 | 能够准确计算折现回收期，比较其与静态回收期、NPV、IRR的优劣，并判断项目决策中的适用性 |
二、我们要解决什么问题？
某公司考虑投资一条新生产线，初始投资800万元，预计未来5年每年产生现金流入，但管理层特别关心“到底要多久才能把钱真正收回来”，而且他们认为“未来的钱不值现在的钱”。静态回收期忽略了货币时间价值，NPV虽然能给出绝对价值，但无法直观告诉管理层“需要几年回本”。折现回收期法正是为了解决这一实际决策痛点：它将所有现金流折现到现值，再计算累计到正值所需的时间，既考虑了时间价值，又保留了回收期“直观易懂”的优点。
三、回收期法回顾与局限性
静态回收期（Payback Period）是最简单的资本预算方法：计算项目累计未折现现金流达到初始投资额所需的时间。
优点：计算简单、直观，适合衡量项目流动性与风险（回收越快，风险越低）。
致命局限：完全忽略货币时间价值，且忽略回收期后的所有现金流，导致可能拒绝NPV为正的优质长期项目。
例如：两个项目初始投资均为100万元，A项目前两年各回收60万元，B项目前三年各回收40万元。静态回收期A为1.67年，B为2.5年，公司可能选A。但若B在第4-6年每年产生50万元现金流，而A之后无现金流，B的NPV可能远高于A。
四、折现回收期（Discounted Payback Period）的定义与计算机制
折现回收期是指将项目各期现金流按要求的资本成本（通常为WACC或要求回报率）折现成现值后，累计这些现值直到等于或超过初始投资额所需的时间。
计算步骤： 1. 确定初始投资额（通常为负值，记为-CF₀）。 2. 确定各期预期现金流CF₁, CF₂, …, CFₙ。 3. 确定折现率r（一般为项目的WACC）。 4. 计算每期现金流的现值：PVₜ = CFₜ / (1+r)ᵗ。 5. 计算累计折现现金流，直至累计值由负转正。 6. 若在第n年转正，则： $$ \text{Discounted Payback} = n-1 + \frac{\text{尚未回收的折现金额}}{\text{第n年的折现现金流}} $$
注意：折现回收期一定大于或等于静态回收期，因为折现使未来现金流变小，需要更长时间才能收回投资。
五、折现回收期与NPV、IRR的比较
- 与NPV的关系：折现回收期本质上是“NPV=0时的时间点”。若项目NPV>0，则折现回收期一定存在且有限；若NPV<0，则折现回收期可能不存在（永远收不回）。
- 与IRR的关系：IRR是使NPV=0的折现率，而折现回收期是使用给定折现率时使累计NPV=0的时间。
- 决策规则：接受折现回收期小于公司设定的最大可接受回收期（Maximum Acceptable Payback）的项目。但该方法仍忽略回收期后的现金流，因此不能单独使用，应与NPV结合判断。
完整案例演算
案例 1：基础计算（等额年金）
某项目初始投资100万元，资本成本10%，未来5年每年产生现金流30万元。
计算过程： - PV₁ = 30 / 1.1 = 27.2727万元 - PV₂ = 30 / 1.21 ≈ 24.7934万元 - PV₃ = 30 / 1.331 ≈ 22.5395万元 - PV₄ = 30 / 1.4641 ≈ 20.4904万元 - PV₅ = 30 / 1.61051 ≈ 18.6277万元
累计折现现金流： - 年末1：-100 + 27.27 = -72.73 - 年末2：-72.73 + 24.79 = -47.94 - 年末3：-47.94 + 22.54 = -25.40 - 年末4：-25.40 + 20.49 = -4.91 - 年末5：-4.91 + 18.63 = +13.72
折现回收期 = 4 + (4.91 / 18.63) ≈ 4 + 0.264 ≈ 4.26年
静态回收期 = 100/30 ≈ 3.33年，可见折现后明显延长。
案例 2：不等额现金流（考试常见情景）
项目初始投资500,000美元，WACC=12%，现金流如下：
| 年份 | 现金流 ($) | 折现系数 (1/1.12ᵗ) | 现值 ($) | 累计现值 ($) |
|---|---|---|---|---|
| 0 | -500,000 | 1.0000 | -500,000 | -500,000 |
| 1 | 150,000 | 0.8929 | 133,935 | -366,065 |
| 2 | 180,000 | 0.7972 | 143,496 | -222,569 |
| 3 | 200,000 | 0.7118 | 142,360 | -80,209 |
| 4 | 160,000 | 0.6355 | 101,680 | +21,471 |
折现回收期 = 3 + (80,209 / 101,680) ≈ 3 + 0.789 ≈ 3.79年
决策：若公司最大可接受折现回收期为4年，则接受该项目。
案例 3：与NPV结合判断（陷阱情景）
两个互斥项目，WACC=10%：
项目A：初始投资200万元，现金流第1-3年各80万元，第4-6年各60万元。
项目B：初始投资200万元，现金流第1-2年各40万元，第3-8年各55万元。
计算结果（略去中间过程）： - 项目A：NPV ≈ +68.4万元，折现回收期 ≈ 3.12年 - 项目B：NPV ≈ +92.7万元，折现回收期 ≈ 3.85年
结论：若仅看折现回收期会选择A，但NPV更高的是B。公司应优先选择B，同时告知管理层虽然B回本稍慢，但总价值更高。这正是考试中最常见的陷阱：回收期法可能与价值最大化冲突。
易错陷阱对照
| 易错点 | 错误做法 | 正确做法 | 考试陷阱 |
|---|---|---|---|
| 混淆静态与折现 | 用未折现现金流累计 | 必须先折现再累计 | 题目故意给出两种结果，让考生选“折现回收期更长” |
| 回收期后现金流 | 认为回收期短的项目一定更好 | 必须结合NPV判断 | 项目A回收快但NPV低，项目B回收慢但NPV高 |
| 折现率选择 | 随意使用IRR | 使用WACC或要求的必要回报率 | 题目给出多个利率，误用IRR计算回收期 |
| 小数点处理 | 直接四舍五入年份 | 必须计算小数部分 | 要求精确到两位小数 |
| 永不回收情况 | 认为NPV<0的项目也有回收期 | 若累计现值永远为负，则无有限回收期 | 题目给出负NPV项目问回收期 |
关键公式 / 关系速记
- 现值：$PV_t = \frac{CF_t}{(1+r)^t}$
- 折现回收期：$n-1 + \frac{\text{剩余未回收现值}}{第n期折现现金流}$
- 关系：Discounted Payback ≥ Payback Period（同一项目）
- 决策规则：若 Discounted Payback < 管理层设定的最大回收期 且 NPV > 0，则接受
- NPV = 0时的折现率 = IRR；累计NPV=0时的时间 = Discounted Payback
练习题（含计算与情景）
Q1. 折现回收期法最主要的改进相对于静态回收期是：
A. 考虑了回收期后的现金流
B. 考虑了货币的时间价值
C. 计算更为简单
D. 能直接给出项目的内部收益率
Q2. 某项目初始投资200万元，WACC=10%，第1年现金流80万元，第2年现金流90万元，第3年现金流100万元。则第2年末累计折现净现金流最接近：
A. -52.89万元
B. -45.12万元
C. -38.43万元
D. -29.75万元
Q3. 下列关于折现回收期的说法正确的是：
A. 它一定小于静态回收期
B. 它忽略了折现率的选择
C. 若项目NPV<0，则通常不存在有限的折现回收期
D. 它比NPV更能全面反映项目价值
Q4. 在计算折现回收期时，折现率通常应采用：
A. 项目内部收益率IRR
B. 加权平均资本成本WACC
C. 无风险利率
D. 预期通胀率
Q5. 项目初始投资1000元，折现率为8%，前三年折现后现金流累计为-180元，第4年折现现金流为320元。该项目的折现回收期为：
A. 3.44年
B. 3.56年
C. 3.63年
D. 4.00年
Q6. 折现回收期法的主要缺点是：
A. 没有考虑货币时间价值
B. 忽略了回收期之后的现金流量
C. 计算过程过于复杂
D. 无法用于互斥项目决策
Q7. 如果两个项目具有相同的初始投资和相同的折现回收期，但NPV不同，则：
A. NPV较高的项目一定具有更长的静态回收期
B. 应优先选择NPV较高的项目
C. 两个项目价值相同
D. 折现回收期更准确，应选择它
Q8. 某项目NPV为负值，则其折现回收期：
A. 必然小于静态回收期
B. 必然等于项目寿命
C. 可能不存在（无限大）
D. 必然等于0
答案与详解
| 题号 | 答案 | 详解 |
|---|---|---|
| Q1 | B | 折现回收期通过将未来现金流折现，明确考虑了货币时间价值，而静态回收期完全不考虑。这是两者最核心的区别。 |
| Q2 | A | PV₁=80/1.1≈72.73；PV₂=90/1.21≈74.38；累计现值=-200+72.73+74.38=-52.89万元。 |
| Q3 | C | 当NPV<0时，累计折现现金流永远无法回正，因此不存在有限的折现回收期。 |
| Q4 | B | 资本预算中折现率一般采用项目的WACC，代表机会成本。 |
| Q5 | B | 3 + 180/320 = 3 + 0.5625 ≈ 3.56年。 |
| Q6 | B | 折现回收期虽然考虑了时间价值，但仍和静态回收期一样，忽略了回收期以后的全部现金流，这是其最主要缺陷。 |
| Q7 | B | 回收期法只是辅助指标，最终决策应以NPV最大化为原则。 |
| Q8 | C | NPV为负意味着即使到项目结束累计现值仍为负，因此折现回收期不存在或为无限大。 |
本节要点速记
- 折现回收期 = 将现金流折现后累计至收回初始投资所需的时间，一定≥静态回收期。
- 计算核心：先折现，再累计，最后用线性插值法求小数年份。
- 决策不能仅依赖回收期，必须与NPV结合使用，否则可能放弃价值更高的长期项目。
- 若NPV<0，通常无有限折现回收期。
- 考试最常见陷阱：给出两个项目，让考生在“回收快但NPV低”与“回收慢但NPV高”之间选择，正确答案永远是NPV优先。
- 公式记忆：$Discounted\ Payback = n-1 + \frac{未回收现值}{n期现值现金流}$。
Corporate Finance
I. Lesson Focus
This lesson explains the discounted payback period as a capital budgeting tool that improves upon the traditional payback period by incorporating the time value of money. Candidates must master its calculation, understand its relationship to NPV and IRR, recognize its advantages and limitations, and know when it should and should not drive project decisions. The focus is on precise computation using uneven cash flows, comparison with static payback, and integration with NPV for correct accept/reject and ranking decisions.
II. The Problem
A company is evaluating a new production line requiring an initial outlay of CNY 8 million and generating cash inflows over five years. Management wants to know exactly how long it will take to recover the investment in today’s dollars, recognizing that future cash is worth less than cash today. The static payback period ignores the time value of money entirely. Although NPV correctly identifies value-creating projects, it does not directly answer the intuitive question “how many years until we get our money back?” The discounted payback period solves this real-world and exam-relevant problem by discounting each cash flow to present value, then measuring the time required for the cumulative discounted cash flows to equal or exceed the initial investment. It balances simplicity with economic realism.
III. Review of Payback Period and Its Limitations
The conventional (static) payback period is the time required for the cumulative undiscounted cash flows to recover the initial investment.
Advantages: Simple to calculate and easy to communicate; useful as a rough liquidity and risk measure (shorter payback implies lower risk).
Critical limitations: Completely ignores the time value of money and discards all cash flows occurring after the payback period. This can lead to rejection of positive-NPV, long-term projects that create substantial value in later years.
Example: Project A recovers CNY 60 million in each of the first two years; Project B recovers CNY 40 million in each of the first three years (both require CNY 100 million initial outlay). Static payback favors A (1.67 years) over B (2.5 years). However, if B generates CNY 50 million annually in years 4–6 while A produces nothing thereafter, B’s NPV will be far superior. Relying solely on static payback would destroy value.
IV. Definition and Calculation Mechanism of Discounted Payback Period
The discounted payback period is the number of years required for the cumulative discounted cash flows to equal or exceed the initial investment. All future cash flows are discounted at the project’s required rate of return (typically WACC).
Calculation Steps: 1. Identify the initial outlay (negative, –CF₀). 2. Forecast periodic cash flows CF₁ to CFₙ. 3. Select the appropriate discount rate r (WACC or hurdle rate). 4. Compute each period’s present value: $PV_t = \frac{CF_t}{(1+r)^t}$. 5. Accumulate the present values until the running total turns from negative to positive. 6. If the sign change occurs in year n, interpolate: $$ \text{Discounted Payback} = n-1 + \frac{\text{Unrecovered discounted amount at end of year }n-1}{PV_n} $$
Important relationship: For any given project, discounted payback is always longer than (or equal to) static payback because discounting reduces the value of future inflows.
V. Discounted Payback versus NPV and IRR
- Link to NPV: Discounted payback identifies the point in time at which cumulative NPV reaches zero. A project with positive NPV will have a finite discounted payback; a negative-NPV project will not.
- Link to IRR: IRR is the discount rate that sets NPV to zero; discounted payback is the time required to set cumulative NPV to zero at a given discount rate.
- Decision Rule: Accept the project if its discounted payback is shorter than the firm’s maximum acceptable payback threshold. However, because it still ignores cash flows beyond the cutoff, it must be used together with NPV. NPV remains the superior criterion for value maximization.
Worked Cases
Case 1: Basic Even Cash Flow (Annuity)
Project requires CNY 1,000,000 initial investment, WACC = 10%, and produces CNY 300,000 annually for five years.
Present Values: - Year 1: 300,000 / 1.10 ≈ 272,727 - Year 2: 300,000 / 1.21 ≈ 247,934 - Year 3: 300,000 / 1.331 ≈ 225,395 - Year 4: 300,000 / 1.4641 ≈ 204,904 - Year 5: 300,000 / 1.61051 ≈ 186,277
Cumulative Discounted Cash Flows (CNY 000s): - End Yr 1: –1,000 + 273 = –727 - End Yr 2: –727 + 248 = –479 - End Yr 3: –479 + 225 = –254 - End Yr 4: –254 + 205 = –49 - End Yr 5: –49 + 186 = +137
Discounted payback = 4 + (49 / 186) ≈ 4 + 0.263 ≈ 4.26 years.
Static payback = 1,000 / 300 ≈ 3.33 years. Discounting lengthens the period, as expected.
Case 2: Uneven Cash Flows (Typical Exam Scenario)
Initial investment USD 500,000, WACC = 12%. Cash flows and calculations:
| Year | Cash Flow ($) | Discount Factor | PV ($) | Cumulative PV ($) |
|---|---|---|---|---|
| 0 | –500,000 | 1.0000 | –500,000 | –500,000 |
| 1 | 150,000 | 0.8929 | 133,935 | –366,065 |
| 2 | 180,000 | 0.7972 | 143,496 | –222,569 |
| 3 | 200,000 | 0.7118 | 142,360 | –80,209 |
| 4 | 160,000 | 0.6355 | 101,680 | +21,471 |
Discounted payback = 3 + (80,209 / 101,680) ≈ 3 + 0.789 = 3.79 years.
If the firm’s maximum acceptable discounted payback is 4.0 years, the project is acceptable.
Case 3: Reconciliation with NPV (Classic Trap)
Two mutually exclusive projects, each requiring CNY 2,000,000, WACC = 10%.
- Project A: CNY 80k in years 1–3, CNY 60k in years 4–6. NPV ≈ +684k, discounted payback ≈ 3.12 years.
- Project B: CNY 40k in years 1–2, CNY 55k in years 3–8. NPV ≈ +927k, discounted payback ≈ 3.85 years.
Conclusion: Using only discounted payback would select A. However, Project B creates substantially more value (higher NPV). The firm should choose B. This conflict between payback ranking and NPV ranking is the most frequent exam trap. NPV must dominate the final decision.
Traps
| Common Mistake | Wrong Approach | Correct Approach | Exam Trap |
|---|---|---|---|
| Confusing static vs. discounted payback | Cumulating undiscounted cash flows | Always discount first, then cumulate | Question supplies both answers; candidates must select the longer discounted figure |
| Ignoring post-payback cash flows | Assuming shorter payback is always superior | Combine with NPV | Project with fast payback has lower NPV; longer-payback project has higher NPV |
| Wrong discount rate | Using IRR to calculate payback | Use WACC | Multiple rates provided; misapplication of IRR produces incorrect payback |
| Rounding error | Rounding to whole years | Retain fractional year with linear interpolation | Questions demand two-decimal precision |
| Negative-NPV projects | Stating a finite payback exists | Recognize that cumulative PV never turns positive | Question asks for payback of a clearly negative-NPV project |
Key Formulas
- Present value of cash flow: $PV_t = \frac{CF_t}{(1+r)^t}$
- Discounted payback period: $n-1 + \frac{\text{Unrecovered PV at end of year }n-1}{PV_n}$
- Relationship: Discounted Payback ≥ Static Payback (same project, same cash flows)
- Decision rule: Accept if Discounted Payback < firm’s maximum acceptable payback and NPV > 0
- NPV = 0 defines IRR; cumulative NPV = 0 defines discounted payback at a fixed r
Practice Questions
Q1. The primary improvement of the discounted payback method over the static payback period is that it:
A. Considers cash flows after the payback period
B. Incorporates the time value of money
C. Is simpler to calculate
D. Directly provides the project’s internal rate of return
Q2. A project requires a CNY 200 million initial investment and has a WACC of 10%. Cash flows are CNY 80m in year 1, CNY 90m in year 2, and CNY 100m in year 3. The cumulative discounted net cash flow at the end of year 2 is closest to:
A. –CNY 52.89 million
B. –CNY 45.12 million
C. –CNY 38.43 million
D. –CNY 29.75 million
Q3. Which statement about the discounted payback period is correct?
A. It is always shorter than the static payback period
B. It ignores the choice of discount rate
C. If a project has negative NPV, it typically has no finite discounted payback
D. It reflects project value more comprehensively than NPV
Q4. When calculating discounted payback, the discount rate should normally be the:
A. Project’s internal rate of return (IRR)
B. Weighted average cost of capital (WACC)
C. Risk-free rate
D. Expected inflation rate
Q5. A project has an initial outlay of 1,000. At an 8% discount rate, the cumulative discounted cash flows through year 3 equal –180. The year-4 discounted cash flow is 320. The discounted payback period is:
A. 3.44 years
B. 3.56 years
C. 3.63 years
D. 4.00 years
Q6. The main disadvantage of the discounted payback method is that it:
A. Ignores the time value of money
B. Ignores all cash flows occurring after the payback period
C. Is excessively complex to compute
D. Cannot be used for mutually exclusive projects
Q7. If two projects have identical initial investments and identical discounted payback periods but different NPVs, an analyst should:
A. Prefer the project with the longer static payback
B. Prefer the project with the higher NPV
C. Consider both projects equally attractive
D. Rely on the payback period as the more accurate metric
Q8. If a project has a negative NPV, its discounted payback period will:
A. Always be shorter than its static payback
B. Equal the project’s economic life
C. Be infinite (or undefined)
D. Equal zero
Answers
| Question | Answer | Explanation |
|---|---|---|
| Q1 | B | Discounted payback explicitly discounts future cash flows, thereby incorporating time value of money. Static payback ignores it completely. |
| Q2 | A | PV₁ = 80 / 1.1 ≈ 72.73; PV₂ = 90 / 1.21 ≈ 74.38; cumulative = –200 + 72.73 + 74.38 = –52.89 million. |
| Q3 | C | Negative NPV implies the cumulative discounted cash flows never turn positive; therefore no finite discounted payback exists. |
| Q4 | B | In capital budgeting the appropriate rate is the project’s WACC, representing the opportunity cost of capital. |
| Q5 | B | 3 + (180 / 320) = 3 + 0.5625 = 3.5625 ≈ 3.56 years. |
| Q6 | B | Although it corrects for time value, discounted payback still discards all cash flows after the cutoff date—the same flaw as static payback. |
| Q7 | B | Payback metrics are supplementary only. NPV is the theoretically superior criterion; choose the higher-NPV project. |
| Q8 | C | A negative NPV means cumulative PV remains negative even at project end; discounted payback is therefore infinite or undefined. |
Takeaways
- Discounted payback = time needed for cumulative discounted cash flows to recover the initial outlay; it is always ≥ static payback.
- Core calculation: discount each cash flow, accumulate, then linearly interpolate the fractional year.
- Never rely on payback alone; always cross-check with NPV. A shorter-payback project can have lower NPV.
- Negative-NPV projects have no finite discounted payback.
- Most frequent exam trap: ranking conflict between payback and NPV—NPV must prevail.
- Formula to remember: $Discounted\ Payback = n-1 + \frac{\text{Unrecovered PV}}{PV_n}$.