Week 7 Note - Capital Budgeting I: Investment Rules, Relevant Cash Flows and Taxation
第 7 周笔记 - 资本预算 I:投资规则、相关现金流和税收
Capital Budgeting - Key Learning Objectives
资本预算 - 关键学习目标
Payback and Discounted Payback:
回收期和折现回收期:
Calculate payback period and discounted payback period.
计算回收期和折现回收期。
Understand the limitations and shortcomings of these methods.
理解这些方法的局限性和不足之处。
Internal Rate of Return (IRR) and Profitability Index (PI):
内部收益率(IRR)和盈利指数(PI):
Compute IRR and PI for investment evaluation.
计算 IRR 和 PI 以进行投资评估。
Recognize the strengths and weaknesses of each approach.
认识到每种方法的优缺点。
Net Present Value (NPV):
净现值(NPV):
Learn to compute NPV.
学会计算 NPV。
Understand why NPV is considered the best decision-making criterion in capital budgeting.
理解为什么 NPV 被认为是资本预算中最佳决策标准。
Relevant Cash Flows (CFs):
相关现金流 (CFs):
Identify and determine the relevant cash flows for assessing a proposed investment.
识别并确定评估拟议投资的相关现金流。
Projected Cash Flows (CFs):
预计现金流 (CFs):
Analyze projected cash flows for evaluating a proposed investment.
分析预计现金流以评估拟议投资。
Net Present Value (NPV):
净现值(NPV):
Estimate NPV and assess investment feasibility based on its value.
根据其价值估算 NPV 并评估投资可行性。
Impact of Inflation:
通货膨胀的影响:
Understand how inflation affects capital budgeting decisions.
了解通货膨胀如何影响资本预算决策。
Impact of Taxation:
税收的影响:
Analyze how taxation influences the viability of investment projects.
分析税收如何影响投资项目的可行性。
Evaluation of Mutually Exclusive Projects:
互斥项目的评估:
Learn to evaluate projects with unequal lifespans and choose the optimal one.
学习评估寿命不等的项目并选择最优项目。
Why Use Net Present Value (NPV)?
为何使用净现值(NPV)?
Shareholder Benefits: Accepting projects with positive NPV increases shareholder value.
股东利益:接受具有正 NPV 的项目可以增加股东价值。
Advantages of NPV:
NPV 的优点:
Uses Cash Flows: Considers actual cash flows rather than accounting profits.
使用现金流:考虑实际现金流而非会计利润。
Comprehensive 全面: Includes all cash flows of the project, ensuring no critical financial aspects are missed.
全面:包括项目的所有现金流,确保没有遗漏关键的财务方面。
Proper Discounting: Appropriately discounts future cash flows to their present value, accurately reflecting the time value of money. 适当地将未来现金流折现为其现值,准确反映货币的时间价值
正确的折现:适当地将未来现金流折现为其现值,准确反映货币的时间价值
The Net Present Value (NPV) Rule
净现值(NPV)规则
Definition:
定义:
NPV = Total PV of future CFs + Initial Investment
NPV = 未来现金流的现值总和 + 初始投资
Steps to Estimate NPV:
估算 NPV 的步骤:
Future Cash Flows: Determine how much cash flows will be generated and when they will occur.
未来现金流:确定将产生多少现金流以及何时发生。
Discount Rate: Estimate the appropriate discount rate to reflect the time value of money and risk.
折现率:估算适当的折现率以反映货币的时间价值和风险。
Initial Costs: Include all upfront investments required for the project.
初始成本:包括项目所需的所有前期投资。
Decision Criteria:
决策标准:
Minimum Acceptance: Accept the project if NPV > 0.
最低接受标准:若净现值(NPV)> 0,则接受该项目。
Ranking: For multiple projects, choose the one with the highest NPV.
排名:对于多个项目,选择 NPV 最高的项目。
The Payback Period Method
回收期法
Disadvantages:
缺点:
Ignore Time Value of Money: Does not discount future cash flows.
忽略货币时间价值:不折现未来现金流。
Ignores Cash Flows After the Payback Period: Overlooks profitability beyond the recovery timeframe.
忽略回收期后的现金流:忽视了回收期外的盈利能力。
Biased Against Long-Term Projects: Favors short-term projects, potentially ignoring better long-term investments.
偏向于短期项目:倾向于短期项目,可能忽略更好的长期投资。
Arbitrary Criteria: The acceptance period is subjective and not grounded in financial logic.
任意标准:接受期主观且缺乏财务逻辑依据。
May Not Ensure Positive NPV: Projects meeting the payback criterion might still destroy value.
可能无法确保正的 NPV:符合回收期标准的项目仍可能摧毁价值。
Advantages:
优点:
Ease of Understanding: Simple concept for stakeholders to grasp.
易于理解:概念简单,便于利益相关者掌握。
Ease of Calculation: Quick and straightforward computation method.
计算简便性:快速且直接的计算方法。
The Discounted Payback Period
折现回收期
Definition:
定义:
Measures how long it takes for a project to "pay back" its initial investment while considering the time value of money.
衡量项目在考虑货币时间价值的情况下收回初始投资所需的时间。
Decision Rule:
决策规则:
Accept the project if it pays back on a discounted basis within the specified time. 如果项目在规定时间内以折扣价返还,则接受该项目
如果项目在规定时间内以折现方式收回,则接受该项目
Key Insight:
关键洞见:
Once cash flows are discounted, you are already calculating values similar to the Net Present Value (NPV). Thus, NPV might provide a more comprehensive evaluation.
一旦现金流量被折现,你实际上已经在计算类似于净现值(NPV)的值。因此,NPV 可能提供更全面的评估。
The Internal Rate of Return (IRR)
内部收益率(IRR)
Definition:
定义:
IRR is the discount rate that makes the Net Present Value (NPV) of a project equal to zero. IRR 是使项目的净现值 (NPV) 等于零的折现率
IRR 是使项目的净现值 (NPV) 等于零的折现率
Decision Criteria:
决策标准:
Minimum Acceptance 最低接受度: Accept the project if the IRR exceeds the required rate of return (hurdle rate). 如果 IRR 超过所需的回报率(最低收益率),则接受该项目
Ranking: Choose the project or alternative with the highest IRR when comparing multiple options. 在比较多个选项时,选择 IRR 最高的项目或替代方案
排名:在比较多个选项时,选择 IRR 最高的项目或替代方案
Reinvestment Assumption:
再投资假设:
Assumes that all future cash flows are reinvested at the IRR, which may not always be realistic. 假设所有未来现金流都以 IRR 进行再投资,但这可能并不总是现实的
假设所有未来现金流都以 IRR 进行再投资,但这可能并不总是现实的
Internal Rate of Return (IRR)
内部收益率(IRR)
Disadvantages:
缺点:
No Distinction Between Investing and Borrowing:
投资与借款无区别:
IRR does not differentiate between cash inflows and outflows.
内部收益率不区分现金流入和流出。
IRR May Not Exist:
内部收益率可能不存在:
For certain projects, a valid IRR cannot be calculated.
对于某些项目,无法计算有效的 IRR。
Multiple IRRs:
多个 IRR:
Projects with unconventional cash flows may have more than one IRR, leading to confusion.
具有非常规现金流的项目可能有多个 IRR,从而导致混淆。
Issues with Mutually Exclusive Investments:
互斥投资的问题:
IRR may not reliably compare mutually exclusive projects.
IRR 可能无法可靠地比较互斥项目。
Advantages:
优点:
Ease of Understanding and Communication:
易于理解和沟通:
IRR is intuitive and easy to explain to stakeholders.
IRR 直观且易于向利益相关者解释。
Example of IRR
IRR 示例
Formula:
公式:
The IRR is the discount rate (IRR) that makes the Net Present Value (NPV) equal to zero:
IRR 是使净现值(NPV)等于零的折现率:
Result:
结果:
Solving the equation gives IRR = 19.44%.
求解该方程得到 IRR 为 19.44%。
Interpretation:
解读:
If the required rate of return (hurdle rate) is less than 19.44%, this project is considered acceptable.
如果要求的回报率(门槛率)小于 19.44%,则该项目被认为是可接受的。
NPV Payoff Profile
NPV 收益曲线
Graph Explanation:
图示说明:
The graph shows the relationship between Net Present Value (NPV) and the discount rate.
该图显示了净现值(NPV)与折现率之间的关系。
As the discount rate increases, the NPV decreases.
随着折现率的增加,NPV 会减少。
Key Observation:
关键观察:
The Internal Rate of Return (IRR) is the discount rate where the NPV equals zero.
内部收益率(IRR)是净现值等于零的折现率。
From the graph, the IRR for this project is 19.44% (the x-axis intercept).
从图中可以看出,该项目的 IRR 为 19.44%(x 轴截距)。
Table Details:
表格详情:
The NPV is calculated for various discount rates (e.g., 0%, 4%, 8%, etc.).
净现值是针对不同的折现率(例如,0%、4%、8%等)计算的。
At 19.44%, the NPV is exactly $0.
19.44%时,净现值正好为 0 美元。
Mutually Exclusive vs. Independent Projects
互斥项目与独立项目
Mutually Exclusive Projects:
互斥项目:
Definition: Only one of several potential projects can be chosen.
定义:在多个潜在项目中只能选择一个。
Example: Choosing between different accounting systems.
示例:在不同的会计系统之间进行选择。
Evaluation Process:
评估过程:
Rank all alternatives based on criteria (e.g., NPV, IRR).
根据标准(例如净现值、内部收益率)对所有备选方案进行排序。
Select the best project from the ranked options.
从排序后的选项中选择最佳项目。
Independent Projects:
独立项目:
Definition: The decision to accept or reject one project does not affect other projects.
定义:接受或拒绝一个项目不会影响其他项目。
Evaluation Criteria:
评估标准:
Each project must meet a minimum acceptance criterion (e.g., NPV > 0 or IRR > required rate).
每个项目必须满足最低接受标准(例如,NPV > 0 或 IRR > 要求的利率)。
Multiple IRRs
多重内部收益率
Project Details:
项目详情:
Initial Investment (Year 0): -$200
初始投资(第 0 年):-$200
Cash Flows:
现金流量:
Year 1: $200
第一年:$200
Year 2: $800
第二年:$800
Year 3: -$800 (negative cash flow).
第三年:-$800(负现金流)。
Key Issue:
关键问题:
The project results in two IRRs:
该项目导致出现两个内部收益率:
IRR₁ = 0%
IRR₂ = 100%
Why Does This Happen?:
为什么会发生这种情况?
Multiple IRRs occur when cash flows change signs more than once (e.g., a mix of positive and negative cash flows).
多次内部收益率(IRR)发生在现金流的正负符号变化超过一次时(例如,正负现金流的混合)。
This creates multiple points where the Net Present Value (NPV) equals zero.
这会导致净现值(NPV)等于零的多个点。
Which IRR to Use?:
使用哪个 IRR?:
The decision is ambiguous when there are multiple IRRs.
当存在多个 IRR 时,决策是模糊的。
Alternative Approach:
另一种方法:
Use the Net Present Value (NPV) method to assess the project, as it provides a clearer and more consistent decision-making framework.
使用净现值(NPV)方法评估该项目,因为它提供了一个更清晰、更一致的投资决策框架。
The Scale Problem
规模问题
Key Question:
关键问题:
Would you prefer a 100% return or a 50% return on your investments?
您更倾向于投资回报率为 100%还是 50%?
Context:
背景:
If the 100% return is on a $1 investment, your profit is $1.
如果 100%的回报是基于 1 美元的投资,您的利润是 1 美元。
If the 50% return is on a $1,000 investment, your profit is $500.
如果 50%的回报是基于 1000 美元的投资,您的利润是 500 美元。
Core Issue:
核心问题:
Scale matters: The percentage return alone does not provide enough information. You need to consider the absolute value of the returns and the investment size to make an informed decision.
规模很重要:仅看百分比回报率并不够。你需要考虑回报的绝对值和投资规模,才能做出明智的决策。
Note: The Net Present Value (NPV) method is better than IRR in such cases because it considers the absolute scale of returns rather than just percentages.
注意:在类似情况下,净现值(NPV)方法比内部收益率(IRR)更好,因为它考虑的是回报的绝对规模,而不仅仅是百分比。
The Timing Problem
时间问题
Core Issue:
核心问题:
Timing of Cash Flows:
现金流的时间:
Project A has earlier returns, while Project B’s returns are mostly in the later years.
项目 A 的回报较早,而项目 B 的回报大多在后期。
IRR may not properly account for the timing differences in cash flows, leading to potential decision-making issues.
内部收益率可能无法正确考虑现金流的时间差异,导致潜在的决策问题。
Resolution:
解决方案:
Use the Net Present Value (NPV) method, which accounts for the time value of money to prioritize projects with better-timed cash flows.
使用净现值(NPV)方法,该方法考虑资金的时间价值,优先考虑现金流更优的项目。
What is the discount rate?
什么是折现率?
In this context, the discount rate refers to the rate used to account for the time value of money. It represents the minimum required return or the cost of capital for the project. Essentially:
在此语境中,折现率是指用于考虑资金时间价值的利率。它代表项目的最低要求回报率或资本成本。本质上:
It adjusts future cash flows to reflect their present value.
它调整未来现金流以反映其现值。
A higher discount rate places less value on future cash flows (e.g., due to higher risk or opportunity cost).
较高的折现率会降低未来现金流的估值(例如,由于风险较高或机会成本增加)。
For example:
例如:
If the discount rate is 10%, $1,000 received in 2 years is worth less today because of the opportunity to invest elsewhere at 10%.
如果折现率为 10%,两年后收到的 1000 美元由于有其他地方可以以 10%的投资机会而今天价值较低。
In project evaluation:
在项目评估中:
Discount rate is often set as the company's cost of capital or a rate reflecting the project's risk.
折现率通常被设定为公司资本成本或反映项目风险的利率。
The Timing Problem – NPV Comparison
时间问题——净现值比较
Graph Analysis:
图形分析:
The chart plots the Net Present Value (NPV) for Project A and Project B against varying discount rates (0% to 40%).
该图表绘制了项目 A 和项目 B 的净现值(NPV)随折现率变化(0%至 40%)的情况。
Observations:
观察:
Project A:
项目 A:
Higher NPV at lower discount rates. 折现率较低时净现值较高
在较低的折现率下净现值较高。 折现率较低时净现值较高
Maintains a positive NPV over a wide range of discount rates. 在各种折现率下均保持正的净现值
在各种折现率下均保持正的净现值。 在各种折现率下均保持正的净现值
More beneficial when the required rate of return is low. 当要求的回报率较低时更有利
当要求的回报率较低时更有利。More beneficial when the required rate of return is low.
Project B:
项目 B:
Outperforms Project A at very high discount rates (e.g., beyond 25%). 在非常高的折现率(例如超过 25%)下表现优于项目 A
在非常高的折现率(例如超过 25%)下表现优于项目 A。Outperforms Project A at very high discount rates (e.g., beyond 25%).
Heavily reliant on later cash flows, making it more sensitive to the discount rate.
严重依赖后期现金流,使其对折现率更敏感。Heavily reliant on later cash flows, making it more sensitive to the discount rate.
Key Insight:
关键洞见:
Project Selection Depends on the Discount Rate:
项目选择取决于折现率:
At lower rates, Project A is preferred due to earlier cash flows.
在较低利率下,由于现金流更早,项目 A 更受青睐。
At higher rates, Project B may be better because of larger, later cash inflows.
在较高利率下,项目 B 可能更好,因为其现金流更大且更晚。
Resolution:
解决方案:
Use NPV as the deciding metric, considering the company’s discount rate or cost of capital to select the optimal project.
使用净现值(NPV)作为决策指标,考虑公司的折现率或资本成本来选择最优项目。
Calculating the Crossover Rate
计算交叉率
Definition:
定义:
The crossover rate is the discount rate at which two projects have the same NPV. 交叉率是两个项目具有相同 NPV 的折现率
交叉率是两个项目具有相同净现值(NPV)的折现率。交叉率是两个项目具有相同 NPV 的折现率
It helps determine when one project becomes more favorable than another. 它有助于确定一个项目何时比另一个项目更有利
它有助于确定一个项目何时比另一个项目更有利。
Graph Explanation:
图示说明:
The crossover rate is the point where the two NPV curves intersect:
交叉点是两个 NPV 曲线相交的点:
At this discount rate, both projects have the same NPV.
在这个折现率下,两个项目的 NPV 相同。
For discount rates below the crossover rate, one project is preferable (e.g., Project A).
对于折现率低于交叉点的,一个项目更可取(例如,项目 A)。
For discount rates above the crossover rate, the other project is better (e.g., Project B).
对于折现率高于交叉点的,另一个项目更好(例如,项目 B)。
Key Insights:
关键洞察:
Use the crossover rate to decide which project is preferable depending on the company's required return or cost of capital.
使用交叉点来决定哪个项目更可取,取决于公司的预期回报或资本成本。
Note: Discount rate below the crossover rate, Project A is better (Additional Info)
注意:折现率低于交叉点率时,项目 A 更优(附加信息)
The reason Project A is better for discount rates below the crossover rate lies in how the timing of cash flows interacts with the discount rate.
项目 A 在折现率低于交叉点率时更优的原因在于现金流的时序与折现率的相互作用。
Key Concept:
关键概念:
Discount Rate: Reflects the time value of money. A lower discount rate places more weight on future cash flows, while a higher discount rate discounts future cash flows more heavily, reducing their present value.
折现率:反映货币的时间价值。较低的折现率对未来的现金流赋予更大的权重,而较高的折现率对未来的现金流进行更大幅度的折现,从而降低其现值。
Crossover Rate: The discount rate where the NPVs of two projects are equal.
交叉点利率:两个项目的净现值相等的折现率。
Comparison of Projects:
项目比较:
Project A:
项目 A:
Has larger early cash flows (e.g., $10,000 in Year 1).
具有较大的早期现金流(例如,第一年 1 万美元)。
At lower discount rates, these early cash flows retain more of their value, making Project A’s NPV higher.
在较低的折现率下,这些早期的现金流保留更多价值,使得项目 A 的净现值更高。
Project B:
项目 B:
Has larger later cash flows (e.g., $12,000 in Year 3).
具有更大的后期现金流(例如,第 3 年为 12,000 美元)。
At higher discount rates, these later cash flows lose value more quickly because they are discounted for more time, making Project A more attractive in comparison.
在较高的折现率下,这些后期现金流因为折现时间更长而更快地失去价值,使得项目 A 相比之下更具吸引力。
Conversely, when discount rates are low, the later cash flows in Project B are not heavily discounted, and its NPV might be closer to or exceed that of Project A.
相反地,当折现率较低时,项目 B 的后期现金流不会被大幅折现,其净现值可能更接近或超过项目 A 的净现值。
Why Project A is Better Below the Crossover Rate:
低于交叉点率时,项目 A 为何更优:
At discount rates below the crossover rate, the early cash flows of Project A have more value compared to Project B’s future-dominated cash flows.
在低于交叉点率的折现率下,项目 A 的早期现金流相比项目 B 未来主导的现金流具有更高价值。
Since Project A delivers its value earlier, it performs better in scenarios where the cost of waiting (time value of money) is low (i.e., lower discount rates).
由于项目 A 更早实现其价值,在等待成本(货币时间价值)较低的情况下(即较低折现率),其表现更优。
In summary:
总结:
Lower discount rate → Less emphasis on time value of money → Early cash flows (Project A) have more relative value.
较低的折现率 → 对货币时间价值的重视程度较低 → 早期现金流(项目 A)具有更高的相对价值。
NPV versus IRR
净现值与内部收益率
General Rule:
一般规则:
NPV (Net Present Value) and IRR (Internal Rate of Return) typically lead to the same decision when evaluating projects.
NPV(净现值)和 IRR(内部收益率)在评估项目时通常会导致相同的决策。
Exceptions:
例外情况:
Non-Conventional Cash Flows:
非传统现金流:
Occurs when cash flow signs change more than once (e.g., negative to positive and back to negative).
当现金流的符号多次变化时发生(例如,从负变正再变负)。
May result in multiple IRRs, causing ambiguity in decision-making.
可能导致多个内部收益率,造成决策上的模糊性。
Mutually Exclusive Projects:
互斥项目:
Initial Investment Differences:
初始投资差异:
When projects have substantially different initial costs, IRR may fail to reflect the better project.
当项目具有显著不同的初始成本时,内部收益率可能无法反映更好的项目。
Timing of Cash Flows:
现金流的时间:
Significant differences in cash flow timing can lead to conflicts between NPV and IRR decisions.
现金流时间的显著差异可能导致 NPV 和 IRR 决策之间的冲突。
The Profitability Index (PI)
盈利指数(PI)
Formula:
公式:
Criteria:
标准:
Minimum Acceptance Criteria:
最低接受标准:
Accept the project if PI > 1.
如果 PI > 1,则接受该项目。
A PI greater than 1 indicates that the project generates more value than its cost.
PI 大于 1 表明该项目产生的价值超过其成本。
Ranking Criteria:
排名标准:
When comparing multiple projects, select the project with the highest PI.
在比较多个项目时,选择投资回收率最高的项目。
The Profitability Index (PI)
盈利指数(PI)
Disadvantages:
缺点:
Mutually Exclusive Investments:
互斥性投资:
PI may not provide the correct decision when projects are mutually exclusive, as it does not account for absolute project sizes or cash flow differences.
PI 在项目互斥时可能无法提供正确决策,因为它不考虑项目的绝对规模或现金流差异。
Advantages:
优点:
Useful for Limited Investment Funds:
适用于有限投资资金:
Helps prioritize projects when resources are constrained, as it identifies projects with the most value per dollar invested.
在资源受限时有助于优先排序项目,因为它能识别每美元投资价值最高的项目。
Easy to Understand and Communicate:
易于理解和沟通:
Simple to explain to stakeholders as it provides a clear ratio of return to cost.
简单向利益相关者解释,因为它提供了清晰的成本回报率。
Effective for Independent Projects:
适用于独立项目:
Provides correct decisions for projects that are independent (i.e., acceptance of one does not affect others).
为独立项目(即一个项目的接受不影响其他项目)提供正确的决策。
Example of Investment Rules: Project A vs. Project B
投资规则的示例:项目 A 与项目 B
Scenario: Compute the IRR, NPV, PI, and payback period for the following two projects. Assume the required return is 10%.
场景:计算以下两个项目的内部收益率、净现值、现值指数和回收期。假设必要回报率为 10%。
Year | Project A | Project B |
0 | -$200 | -$150 |
1 | $200 | $50 |
2 | $800 | $100 |
3 | -$800 | $150 |
Summary of Metrics:
指标总结:
Metric | Project A | Project B |
Initial Investment (CF₀) | -$200 | -$150 |
Present Value (PV₀ of CF₁-₃) | $200 | $50 |
Net Present Value (NPV) | $800 | $100 |
Internal Rate of Return (IRR) | 0%, 100% | 36.19% |
Profitability Index (PI) | 1.2096 | 1.6053 |
Key Insights:
关键洞察:
Net Present Value (NPV):
净现值(NPV):
Project B has a higher NPV ($90.80) compared to Project A ($41.92), indicating it creates more value.
项目 B 的 NPV(90.80)高于项目 A(41.92),表明其创造了更多价值。
Internal Rate of Return (IRR):
内部收益率 (IRR):
Project A has two IRRs (0% and 100%) due to its non-conventional cash flows (sign changes).
项目 A 有两个内部收益率(0%和 100%),这是由于其非传统现金流(符号变化)。
Project B has a single IRR of 36.19%, which exceeds the required return of 10%.
项目 B 有一个内部收益率为 36.19%,超过了要求的回报率 10%。
Profitability Index (PI):
盈利能力指数 (PI):
Project B has a higher PI (1.6053), making it a better option if resources are limited.
项目 B 具有更高的 PI(1.6053),如果资源有限,则它是更好的选择。
Payback Period Analysis
回收期分析
Cumulative Cash Flows:
累计现金流:
Year | CF (Project A) | Cum. CF (Project A) | Project B | Cum. CF (Project B) |
0 | -$200 | -$200 | -$150 | -$150 |
1 | $200 | 0 | $50 | -$100 |
2 | $800 | $800 | $100 | 0 |
3 | -$800 | 0 | $150 | $150 |
Payback Period:
回收期:
Project B:
项目 B:
The cumulative cash flow becomes positive at 2 years, so the payback period is 2 years.
累计现金流在 2 年时变为正数,因此回收期为 2 年。
Project A:
项目 A:
The cumulative cash flow reaches $0 twice:
累计现金流达到 0 两次:
At the end of Year 1.
在第一年年底。
At the end of Year 3, due to the negative cash flow in Year 3.
在第三年年底,由于第三年出现了负现金流。
Ambiguity: The payback period depends on how the method handles negative cash flows after initial recovery:
模糊性:回收期取决于该方法如何处理初始回收后的负现金流:
If no further consideration of negative cash flows: Payback is 1 year
如果不再考虑负现金流:回收期为 1 年.
If full repayment is required: Payback is 3 years.
如果需要全额还款:回收期为 3 年。
Key Insight:
关键洞见:
Project B has a clearer payback period (2 years).
项目 B 具有更清晰的回收期(2 年)。
Project A demonstrates why the payback period is less reliable for projects with fluctuating cash flows.
项目 A 展示了为什么回收期对于现金流波动的项目来说不太可靠。
NPV and IRR Relationship
NPV 与 IRR 的关系
Summary of NPV Values at Different Discount Rates:
不同折现率下的 NPV 值总结:
Discount Rate | NPV for A | NPV for B |
-10% | -87.52 | 234.77 |
0% | 0.00 | 150.00 |
20% | 59.26 | 47.92 |
40% | 59.48 | -8.60 |
60% | 42.19 | -43.07 |
80% | 20.85 | -65.64 |
100% | 0.00 | -81.25 |
120% | -18.93 | -92.52 |
Key Insights:
关键洞察:
IRR and NPV Relationship:
内部收益率与净现值的关系:
For Project A:
对于项目 A:
NPV equals $0 at 0% and 100% discount rates, indicating two IRRs due to the non-conventional cash flows.
净现值在 0%和 100%的折现率下等于 0,这表明由于非传统现金流,存在两个内部收益率。
For Project B:
对于项目 B:
NPV equals $0 at approximately 36.19% (the single IRR for B).
净现值在约 36.19%(B 的唯一内部收益率)时等于 0。
Comparison at Discount Rates:
不同折现率下的比较:
At low discount rates (e.g., 0%), Project B has a higher NPV, making it more favorable.
在低折现率(例如 0%)时,项目 B 的净现值更高,因此更有利。
At high discount rates (e.g., 40% or more), Project A consistently performs better because its cash flows are less sensitive to the discount rate.
在高折现率(例如 40%或更高)时,项目 A 始终表现更好,因为其现金流对折现率的敏感度较低。
Crossover Point:
交叉点:
Between 20% and 40%, the NPV for both projects changes, which may indicate the crossover rate where their NPVs are equal.
在 20%到 40%之间,两个项目的净现值发生变化,这可能表明它们的净现值相等的交叉点。
Conclusion:
结论:
The choice between projects depends on the discount rate.
项目选择取决于折现率。
At low discount rates, Project B is preferred; at higher discount rates, Project A becomes the better option.
在低折现率下,项目 B 更受青睐;在高折现率下,项目 A 成为更好的选择。