📚 Worksheet Answers – Chapter 26 Investment Appraisal | 第26章 投资评估 工作表答案
This worksheet covers Chapter 26 on investment appraisal for Cambridge A-Level Business. It provides model answers for payback period, average rate of return (ARR), net present value (NPV), and qualitative factors in capital investment decisions. Use these answers to check your calculations and improve your exam technique.
本工作表答案对应剑桥A-Level商务第26章“投资评估”。内容包括回收期、平均收益率、净现值和资本投资决策中的定性因素。请使用这些参考答案核对你的计算，并提升考试答题技巧。
1. Payback Period for Project Alpha | 项目Alpha的回收期
Project Alpha has an initial cost of $100,000 and expected net cash inflows of $30,000 in Year 1, $40,000 in Year 2, $50,000 in Year 3, $20,000 in Year 4, and $10,000 in Year 5. The cumulative cash flow after Year 2 is $70,000, so an additional $30,000 is needed from Year 3.
项目Alpha的初始成本为100,000美元，预计净现金流入为：第1年30,000美元、第2年40,000美元、第3年50,000美元、第4年20,000美元、第5年10,000美元。第2年末累计现金流为70,000美元，因此还需要从第3年获得30,000美元。
The payback period is therefore 2 years plus the fraction of Year 3 needed to recover the remaining $30,000 out of the $50,000 received in Year 3.
因此，回收期为2年加上第3年收回剩余30,000美元所需的时间，第3年共收到50,000美元。
Payback period = 2 + (100,000 − 70,000) ÷ 50,000 = 2.6 years
Project Alpha pays back its initial investment in 2.6 years. This is below the company’s target payback period of 3 years, so the project is acceptable on this criterion.
项目Alpha在2.6年内收回初始投资。该回收期低于公司设定的3年目标回收期，因此该项目在此标准下是可行的。
2. Payback Period for Project Beta with Uneven Cash Flows | 项目Beta非均匀现金流下的回收期
Project Beta costs $120,000 and has net cash inflows of $20,000 in Year 1, $30,000 in Year 2, $40,000 in Year 3, $60,000 in Year 4, and $30,000 in Year 5. By the end of Year 3, cumulative cash flow is $90,000.
项目Beta成本为120,000美元，净现金流入为：第1年20,000美元、第2年30,000美元、第3年40,000美元、第4年60,000美元、第5年30,000美元。到第3年末，累计现金流为90,000美元。
An additional $30,000 is needed from Year 4, which provides $60,000. The payback calculation is shown below.
还需从第4年获得30,000美元，而第4年提供60,000美元。回收期计算如下。
Payback period = 3 + (120,000 − 90,000) ÷ 60,000 = 3.5 years
Project Beta has a payback period of 3.5 years. Since this exceeds the target of 3 years, Beta would be rejected if payback were the only decision rule.
项目Beta的回收期为3.5年。由于该回收期超过3年目标回收期，如果只使用回收期法进行决策，Beta项目将被拒绝。
3. Average Rate of Return for Project Alpha | 项目Alpha的平均收益率
The average rate of return (ARR) measures the average annual profit as a percentage of the initial investment. Total cash inflows for Alpha are $150,000, and total profit over five years is $150,000 − $100,000 = $50,000.
平均收益率衡量的是平均年利润占初始投资的百分比。项目Alpha的总现金流入为150,000美元，五年总利润为150,000 − 100,000 = 50,000美元。
Average annual profit is $50,000 ÷ 5 = $10,000. Using the initial investment method:
平均年利润为50,000 ÷ 5 = 10,000美元。按初始投资法计算：
ARR = 10,000 ÷ 100,000 × 100 = 10%
Project Alpha has an ARR of 10%. If the company requires a minimum ARR of 12%, Alpha would be rejected on this measure. If using average investment, the denominator becomes (100,000 + 0) ÷ 2 = 50,000, giving an ARR of 20%, which is higher.
项目Alpha的平均收益率为10%。如果公司要求最低平均收益率为12%，则按此指标Alpha将被拒绝。若使用平均投资额法，分母为(100,000 + 0) ÷ 2 = 50,000，平均收益率为20%，会更高。
4. Average Rate of Return for Project Beta | 项目Beta的平均收益率
Project Beta has total cash inflows of $180,000 and an initial cost of $120,000. Total profit over five years is $180,000 − $120,000 = $60,000.
项目Beta的总现金流入为180,000美元，初始成本为120,000美元。五年总利润为180,000 − 120,000 = 60,000美元。
Average annual profit is $60,000 ÷ 5 = $12,000. Using the initial investment method:
平均年利润为60,000 ÷ 5 = 12,000美元。按初始投资法计算：
ARR = 12,000 ÷ 120,000 × 100 = 10%
Both Alpha and Beta have the same ARR of 10% under the initial investment method. This shows that ARR alone does not always distinguish between projects, especially when project sizes differ.
在初始投资法下，Alpha和Beta的平均收益率均为10%。这说明平均收益率法并不总能区分不同项目，尤其是当项目规模不同时。
5. Net Present Value for Project Alpha at 10% | 项目Alpha在10%折现率下的净现值
Net present value discounts future cash flows to their present value and subtracts the initial investment. The discount factors at 10% for Years 1 to 5 are 0.909, 0.826, 0.751, 0.683, and 0.621.
净现值将未来现金流折现为现值并减去初始投资。10%折现率下第1至第5年的折现系数分别为0.909、0.826、0.751、0.683和0.621。
| Year | Cash inflow | Discount factor at 10% | Present value |
|---|---|---|---|
| 1 | $30,000 | 0.909 | $27,270 |
| 2 | $40,000 | 0.826 | $33,040 |
| 3 | $50,000 | 0.751 | $37,550 |
| 4 | $20,000 | 0.683 | $13,660 |
| 5 | $10,000 | 0.621 | $6,210 |
Total present value of inflows is $117,730. Subtracting the initial cost of $100,000 gives:
现金流入总现值为117,730美元。减去初始成本100,000美元后：
NPV = 117,730 − 100,000 = $17,730
Alpha has a positive NPV of $17,730 at a 10% discount rate. A positive NPV means the project earns more than the required rate of return and should be accepted.
Alpha在10%折现率下的净现值为17,730美元，为正值。净现值为正意味着项目收益超过要求回报率，应当接受该项目。
6. Net Present Value for Project Beta at 10% | 项目Beta在10%折现率下的净现值
Using the same 10% discount factors, the present values for Beta’s cash inflows are calculated as follows.
使用相同的10%折现系数，项目Beta现金流入的现值计算如下。
| Year | Cash inflow | Discount factor | Present value |
|---|---|---|---|
| 1 | $20,000 | 0.909 | $18,180 |
| 2 | $30,000 | 0.826 | $24,780 |
| 3 | $40,000 | 0.751 | $30,040 |
| 4 | $60,000 | 0.683 | $40,980 |
| 5 | $30,000 | 0.621 | $18,630 |
Total present value of inflows is $132,610. Subtracting the initial cost of $120,000 gives:
现金流入总现值为132,610美元。减去初始成本120,000美元后：
NPV = 132,610 − 120,000 = $12,610
Beta also has a positive NPV of $12,610. Both projects add value, but Alpha has the higher NPV, so Alpha is preferred under the NPV rule.
Beta的净现值也为正值，为12,610美元。两个项目都能增加价值，但Alpha的净现值更高，因此根据净现值法应优先选择Alpha。
7. Payback and NPV Decision Rules | 回收期与净现值决策规则
The payback decision rule accepts a project only if its payback period is less than or equal to a target period set by management. In this worksheet the target is 3 years.
回收期决策规则只有当项目回收期小于或等于管理层设定的目标回收期时才接受项目。本工作表中目标回收期为3年。
The NPV rule accepts a project if its NPV is positive at the chosen discount rate. A positive NPV means the project increases shareholder wealth because it earns a return above the cost of capital.
净现值规则是在所选折现率下，如果净现值为正就接受项目。净现值为正意味着项目增加股东财富，因为其回报率高于资本成本。
For Alpha and Beta, payback gives a different ranking from NPV. Alpha has a shorter payback and a higher NPV, so both methods favour Alpha over Beta.
对于Alpha和Beta，回收期法与净现值法给出了不同的排序。Alpha的回收期更短且净现值更高，因此两种方法都更倾向于Alpha而非Beta。
8. Limitations of Payback and ARR | 回收期与平均收益率法的局限性
The payback method ignores the time value of money, because it treats cash received in Year 1 the same as cash received in Year 3. It also ignores cash flows after the payback period, which can lead to poor long-term decisions.
回收期法忽略了货币时间价值，因为它将第1年收到的现金与第3年收到的现金同等对待。它还忽略了回收期之后的现金流，这可能导致长期决策失误。
ARR also ignores the time value of money because it uses average accounting profit rather than discounted cash flows. It can be calculated in different ways, which makes comparison between businesses difficult.
平均收益率法同样忽略货币时间价值，因为它使用的是平均会计利润而非折现现金流。它的计算方法有多种，这使得企业间的比较更加困难。
Both payback and ARR can be useful as quick screening tools, but they should not be used alone for major capital investment decisions.
回收期法和平均收益率法都可以作为快速筛选工具，但在重大资本投资决策中不应单独使用。
9. Qualitative Factors in Investment Appraisal | 投资评估中的定性因素
Quantitative methods such as payback, ARR, and NPV do not capture all relevant factors. Managers must also consider qualitative issues before making a final investment decision.
回收期、平均收益率和净现值等定量方法并未涵盖所有相关因素。管理者在作出最终投资决策前还必须考虑定性因素。
- Risk and uncertainty: future cash flows may be less certain than the forecasts suggest.
- Strategic fit: the project should support the business’s long-term objectives and brand image.
- Environmental and social impact: negative externalities may damage reputation or lead to regulation.
- Managerial experience and capacity: the management team must be able to run the new project effectively.
- Flexibility: some projects allow expansion or exit options, which can add value.
以上要点中文翻译：
- 风险与不确定性：未来现金流可能不如预测中那么确定。
- 战略匹配：项目应支持企业的长期目标和品牌形象。
- 环境与社会影响：负外部性可能损害声誉或招致监管。
- 管理经验与能力：管理团队必须能够有效运营新项目。
- 灵活性：某些项目具有扩张或退出期权，这可以增加价值。
10. Final Recommendation | 最终建议
Based on payback period, Alpha is clearly preferred because 2.6 years is below the 3-year target, while Beta takes 3.5 years. Based on NPV, Alpha also has a higher positive NPV of $17,730 compared with Beta’s $12,610.
根据回收期，Alpha明显更优，因为2.6年低于3年目标，而Beta需要3.5年。根据净现值，Alpha同样更优，其净现值为17,730美元，高于Beta的12,610美元。
ARR gives the same result of 10% for both projects under the initial investment method, so it does not change the recommendation. If the company has limited funds, Alpha should be selected first.
平均收益率法在初始投资法下两个项目的结果都是10%，因此不会改变建议。如果公司资金有限，应首先选择Alpha项目。
However, if Beta offers stronger strategic advantages, such as entering a new market or reducing long-term risk, managers may still choose Beta despite its lower NPV and longer payback. The final decision should combine quantitative results with qualitative judgement.
但是，如果Beta具有更强的战略优势，例如进入新市场或降低长期风险，管理者仍可能选择Beta，尽管其净现值较低且回收期较长。最终决策应将定量结果与定性判断相结合。
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