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%。

Summary of Metrics:
指标总结：

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：

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：

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。

NPV Profiles for Project A and Project B
项目 A 和项目 B 的净现值曲线图

Interpretation:
解读：

Below the Crossover Rate:
低于交叉点利率：

Project B is preferred due to higher NPV.
由于 NPV 更高，项目 B 更受青睐。

Above the Crossover Rate:
高于交叉点利率：

Project A becomes more favorable as its earlier cash flows are less affected by high discount rates.
项目 A 随着其早期现金流受高折现率的影响减小而变得更加有利。

Conclusion:
结论：

Use the crossover rate and required return (discount rate) to determine the optimal project.
使用交叉点和要求回报（折现率）来确定最佳项目。

When in doubt, rely on NPV as the decision metric.
不确定时，依赖净现值作为决策指标。

Where Do the Cash Flows (CFs) Come From?
现金流量（CFs）从哪里来？

Sales and Other Revenues:
销售收入和其他收入：

The inflows generated from selling goods or services and other revenue streams.
来自销售商品或服务及其他收入来源的现金流入。

Cost of Sales and Expenses:
销售成本和费用：

Includes the direct costs of producing goods or services and operating expenses that reduce cash inflows.
包括生产商品或服务产生的直接成本以及减少现金流入的运营费用。

Depreciation:
折旧：

A non-cash expense that affects taxable income and, indirectly, cash flows through tax savings (depreciation tax shield).
一种非现金费用，通过税收节省（折旧税盾）间接影响应税收入和现金流量。

Working Capital:
营运资金：

Changes in current assets and liabilities (e.g., inventory, receivables, payables) that impact cash availability.
流动资产和负债的变化（例如库存、应收账款、应付账款）对现金流量的影响。

Taxes:
税收：

Cash outflows or savings related to corporate tax obligations, influenced by taxable income and deductions.
与公司税收义务相关的现金流出或节约，受应税收入和扣除额的影响。

Revenues and Cash Flows (CFs)
收入和现金流（CFs）

Impact of Revenues on CFs:
收入对现金流的影响：

Revenues contribute positively to cash flows by a factor of (1 − ${''t''}_{\mathrm{}}$), where:
收入通过(1 − ${''t''}_{\mathrm{}}$ )的系数对现金流产生正向影响，其中：

${''t''}_{\mathrm{}}$ = company tax rate.
${''t''}_{\mathrm{}}$ = 公司税率。

Effect of Changes in Revenue:
收入变化的影响：

Increase in Revenue: Each $1 increase in revenue results in an increase in CF by 1 − ${''t''}_{\mathrm{}}$.
收入增加：每增加 1 美元收入，CF 将增加 1 − ${''t''}_{\mathrm{}}$ 。

Decrease in Revenue: Each $1 decrease in revenue reduces CF by 1 − ${''t''}_{\mathrm{}}$.
收入减少：每减少 1 美元收入，CF 减少 1 − ${''t''}_{\mathrm{}}$ 。

Example:
示例：

If ${''t''}_{\mathrm{}}$ =30% (tax rate):
如果 ${''t''}_{\mathrm{}}$ =30%（税率）：

A $1 increase in revenue adds $0.70 to cash flows.
收入增加 1 美元，现金流增加 0.70 美元。

A $1 decrease in revenue reduces cash flows by $0.70.
收入减少 1 美元，现金流减少 0.70 美元。

Costs and Cash Flows (CFs)
成本与现金流（CFs）

Impact of Costs on CFs:
成本对现金流的影响：

Costs contribute negatively to cash flows by a factor of (1 − ${''t''}_{\mathrm{}}$), where:
成本通过(1 − ${''t''}_{\mathrm{}}$ )的系数对现金流产生负面影响，其中：

${''t''}_{\mathrm{}}$ = company tax rate.
${''t''}_{\mathrm{}}$ = 公司税率。

Effect of Changes in Costs:
成本变化的影响：

Increase in Costs: Each $1 increase in costs results in a decrease in CF by 1 − ${''t''}_{\mathrm{}}$.
成本增加：每增加 1 美元成本，导致 CF 减少 1 − ${''t''}_{\mathrm{}}$ 。

Decrease in Costs: Each $1 decrease in costs increases CF by 1 − ${''t''}_{\mathrm{}}$.
成本减少：每减少 1 美元成本，CF 增加 1 − ${''t''}_{\mathrm{}}$ 。

Key Adjustment:
关键调整：

All real cash flows are multiplied by (1 − ${''t''}_{\mathrm{}}$) to reflect their after-tax impact.
所有实际现金流都乘以（1 − ${''t''}_{\mathrm{}}$ ）以反映其税后影响。

Example:
示例：

If ${''t''}_{\mathrm{}}$ =30% (tax rate):
如果 ${''t''}_{\mathrm{}}$ =30%（税率）：

A $1 increase in costs reduces CF by $0.70.
成本增加 1 美元将使现金流量减少 0.70 美元。

A $1 decrease in costs increases CF by $0.70.
成本减少 1 美元将使现金流量增加 0.70 美元。

Note: Information
注意：信息

A $1 increase in costs reduces cash flow by $0.70 because costs are tax-deductible, meaning they reduce taxable income, which reduces the tax payable. Here's how it works:
一项成本增加 1 美元会减少 0.70 美元的现金流，因为成本是可抵扣税的，这意味着它们会减少应税收入，从而减少应纳税额。具体工作原理如下：

Tax Deduction:
税收抵扣：

For every $1 increase in costs, taxable income decreases by $1.
每增加 1 美元的成本，应税收入就会减少 1 美元。

This results in a tax savings equal to ${''t''}_{\mathrm{}}$, the tax rate. For example, if the tax rate (${''t''}_{\mathrm{}}$) is 30%, the tax savings is $0.30.
这会导致相当于税率的税收节省，即 ${''t''}_{\mathrm{}}$ 。例如，如果税率（ ${''t''}_{\mathrm{}}$ ）为 30%，税收节省就是 0.30 美元。

Net Effect:
净效应：

The $1 increase in costs reduces cash flow by the full $1.
成本增加 1 美元，使现金流减少 1 美元。

However, you save $0.30 in taxes, so the net impact on cash flow is 1 - 0.30 = $0.70.
然而，你节省了 0.30 美元的税款，因此对现金流的影响是 1 - 0.30 = 0.70 美元。

Conclusion:
结论：

The reduction in cash flow is less than the full $1 because part of the cost is offset by the tax savings.
现金流的减少额小于全部 1 美元，因为部分成本被税收节省所抵消。

Depreciation and Cash Flows (CFs)
折旧与现金流（CFs）

Depreciation is Not a Real CF:
折旧不是真实的现金流：

Depreciation is a non-cash expense that does not directly involve cash outflows.
折旧是一项非现金支出，不直接涉及现金流出。

Importance of Depreciation:
折旧的重要性：

It provides a tax shelter by reducing taxable income, leading to tax savings.
它通过减少应税收入提供税收庇护，从而节省税款。

These tax savings effectively increase cash flow.
这些税收节省有效地增加了现金流。

Impact on Cash Flows:
对现金流的影响：

Positive Effect: Depreciation reduces taxes owed, which increases net cash flows.
正面效应：折旧减少应缴税款，从而增加净现金流。

Every $1 increase (decrease) in depreciation results in an increase (decrease) in cash flow by ${''t''}_{\mathrm{}}$ (the tax rate).
每增加（减少）1 美元的折旧，现金流量将增加（减少） ${''t''}_{\mathrm{}}$ （税率）。

Example:
示例：

If ${''t''}_{\mathrm{}}$ =30%, a $1 increase in depreciation saves $0.30 in taxes, which is added to cash flow.
如果 ${''t''}_{\mathrm{}}$ =30%，折旧增加 1 美元将节省 0.30 美元的税款，这会增加到现金流量中。

Depreciation
折旧

Definition:
定义：

Depreciation expense is the amount allowed by tax authorities (e.g., ATO) for tax purposes.
折旧费用是税务机关（例如 ATO）为税务目的允许的金额。

It represents the allocation of the cost of an asset over its useful life.
它表示资产成本在其使用寿命内的分配。

Non-Cash Nature:
非现金性质：

Depreciation is a non-cash expense and does not involve actual cash outflows.
折旧是一种非现金费用，不涉及实际的现金支出。

Its relevance lies in its effect on taxes, as it reduces taxable income (the company pays less in taxes).
其相关性在于其对税收的影响，因为它减少了应税收入（公司缴纳的税款减少）。

Depreciation Tax Shield:
折旧税盾：

The tax savings due to depreciation is called the depreciation tax shield.
由于折旧产生的税收节省被称为折旧税盾。

Formula:
公式：

Key Impact:
主要影响：

The depreciation tax shield increases cash flow by reducing taxes owed.
折旧税盾通过减少应缴税款来增加现金流。

Example:
示例：

Computing Depreciation
计算折旧

Prime Cost (Straight Line) Method:
成本法（直线法）：

Definition:
定义：

A constant depreciation rate (r) is applied to the asset's initial cost or book value at time t=0 (${''BV''}_{\mathrm{}}$).
一个恒定的折旧率（r）应用于资产在 t=0（ ${''BV''}_{\mathrm{}}$ ）时的初始成本或账面价值。

This method results in a fixed depreciation amount (D) each year.
这种方法每年导致折旧金额（D）固定。

Formula:
公式：

D = r × ${''BV''}_{\mathrm{}}$

D: Annual depreciation expense.
D: 年度折旧费用。

r: Depreciation rate (a constant percentage).
r: 折旧率（一个固定的百分比）。

${''BV''}_{\mathrm{}}$: Initial book value of the asset.
${''BV''}_{\mathrm{}}$ ：资产的初始账面价值。

Key Characteristics:
关键特征：

The depreciation amount is the same each year over the asset's useful life.
折旧金额在资产的整个使用年限内每年都相同。

Suitable for assets that lose value evenly over time.
适用于随时间均匀贬值的资产。

Example:
示例：

If an asset's initial cost (${''BV''}_{\mathrm{}}$) is $50,000 and the depreciation rate (r) is 10%:
如果一项资产的初始成本（ ${''BV''}_{\mathrm{}}$ ）为 50,000 美元，折旧率（r）为 10%：

D=0.10×50,000=5,000

Annual depreciation is $5,000.
年折旧额为 5,000 美元。

Example 1: After-Tax Cash Flow Change
示例 1：税后现金流变化

Suppose Splash Ltd plans to replace equipment which will result in additional revenues of $9,000 but also increased costs by $4,000 per annum
假设 Splash Ltd 计划更换设备，这将导致额外收入 9,000 美元，但同时每年增加成本 4,000 美元

Assume that Splash is able to claim additional depreciation of $3,000
假设 Splash 能够额外折旧 3,000 美元

By how much will its after-tax cash flow change if the company tax rate, ${''t''}_{\mathrm{}}$ is 30%?
如果公司税率 ${''t''}_{\mathrm{}}$ 为 30%，其税后现金流将变化多少？

Answer-Example 1: After-Tax Cash Flow (Method 1)
答案示例 1：税后现金流（方法 1）

Formula:
公式：

Given Values:
给定值：

Revenues (R): $9,000.
收入（R）：9,000 美元。

Costs (C): $4,000.
成本（C）：4,000 美元。

Tax rate (${''t''}_{\mathrm{}}$): 30%.
税率 ( ${''t''}_{\mathrm{}}$ ): 30%。

Depreciation: $3,000.
折旧: 3,000 美元。

Answer- Example 1: After-Tax Cash Flow (Method 2)
答案-示例 1：税后现金流（方法 2）

Formula:
公式：

Answer- Example 1: After-Tax Cash Flow (Method 3)
答案-示例 1：税后现金流（方法 3）

Answer- Example 1: After-Tax Cash Flow (Method 4)
答案-示例 1：税后现金流（方法 4）

Note: Only need to know one method
注意：只需知道一种方法

Net Working Capital (WC)
净营运资本 (WC)

Definition:
定义：

WC represents cash employed to manage the day-to-day operations of a firm, such as investments in inventory or receivables. WC 代表用于管理公司日常运营的现金，例如库存或应收账款的投资
WC 代表用于管理公司日常运营的现金，例如库存或应收账款投资。 WC 代表管理公司日常运营的现金，例如库存或应收账款的投资

Nature:
性质：

Not consumed: WC is not permanently spent but is employed temporarily for operations.
未消耗：营运资金并非永久性支出，而是临时用于运营的资金。

Impact of WC Changes:
营运资金变化的影响：

Increase in WC:
营运资本增加：

More cash is tied up (e.g., higher inventory or receivables).
更多的现金被占用（例如，更高的库存或应收账款）。

Represents a cash outflow.
代表现金流出。

Decrease in WC:
营运资本减少：

Less cash is tied up (e.g., reduced inventory or receivables).
占用的现金更少（例如，减少库存或应收账款）。

Represents a cash inflow.
表示现金流入。

Key Point:
要点：

WC adjustments affect cash flows during the operating cycle but are generally recovered at the end of the project or period.
营运资本调整会影响经营周期内的现金流，但通常在项目或期末得到回收。

Taxation
税收

Three Major Impacts:
三大主要影响：

Income Tax 所得税:
所得税:

Represents a cash outflow, reducing the net cash flow available to the business.
代表现金流出，减少可用于企业的净现金流。

Tax Shield 税盾:
税盾 Tax Shield:

Depreciation provides a tax deduction.
折旧提供税收抵免。

Results in tax savings by lowering taxable income.
通过降低应税收入来节省税款。

Capital Gains Tax (CGT) 资本利得税:
资本利得税 (CGT):

Effect on Net Profit:
对净利润的影响:

Reduces the profit earned from the sale of an asset.
降低了资产出售所获得的利润。

Tax Savings on Loss:
亏损时的税收节省

If a loss is incurred from the sale of an asset, it may result in tax savings.
如果资产出售发生亏损，可能会导致税收节省。

Disposal of an Asset
资产处置

General Outcomes:
一般结果：

Gain/Profit: Occurs when the asset is sold for more than its book value.
收益/利润：当资产出售价格高于其账面价值时发生。

Loss: Occurs when the asset is sold for less than its book value.
损失：当资产以低于其账面价值的价格出售时发生。

Taxation Implications:
税收影响：

Gain: Subject to tax.
收益：需缴纳税收。

Loss: Leads to a tax deduction, reducing the amount of tax paid.
损失：可抵扣税收，减少应缴税款。

What to Discount
要折现什么

General Rules 一般规则:
一般规则:

Relevance of Cash Flow: Focus only on cash flows, disregarding accounting income. 现金流相关性：只关注现金流，不考虑会计收入
现金流的相关性：只关注现金流，不考虑会计收入

Incremental Basis: Always estimate cash flows as the difference brought about by a decision or change. 增量基础：始终将现金流估算为决策或变更带来的差异
增量基础：始终将现金流量估计为决策或变更带来的差异。 增量基础：始终将提示作为决策或变更带来的差异

Consistency with Inflation: Ensure uniform treatment of inflation across all calculations. 与通货膨胀保持一致：确保在所有计算中统一处理通货膨胀
与通货膨胀保持一致：确保在所有计算中统一处理通货膨胀

Note: “discount" refers to the process of adjusting future cash flows to account for the time value of money. It means reducing the value of future cash flows to reflect their worth in today's terms, using a specified discount rate.
注意：“折现”是指调整未来现金流以考虑货币时间价值的过程。它意味着使用特定的折现率降低未来现金流的价值，以反映它们在今天的价值。

Common Types of Cash Flows (CFs) to Watch Out For
需要关注的常见现金流类型

Sunk Costs: Costs incurred in the past and irrelevant to future decisions.
沉没成本：过去发生的与未来决策无关的成本。

Opportunity Costs: Value of benefits lost from the next best alternative.
机会成本：因选择次优方案而失去的收益价值。

Side Effects (Externalities):
副作用（外部性）：

Positive: Benefits to other projects.
正面影响：对其他项目的收益。

Negative: Costs to other projects.
负面影响：对其他项目的成本。

Changes in Net Working Capital (NWC): Fluctuations in working capital leading to cash inflows/outflows.
净营运资本（NWC）的变化：营运资本波动导致的现金流入/流出。

Financing Costs: Accounted for in the discount rate and not separately in CF calculations.
融资成本：计入折现率中，不在现金流量计算中单独体现。

Taxes: Consider tax implications and adjustments in cash flows.
税收：考虑税收影响和现金流量的调整。

Ignore Sunk Costs
忽略沉没成本

Definition: Sunk costs are past cash outflows that cannot be recovered and are irrelevant to current or future decisions.
定义：沉没成本是指过去无法收回的现金支出，与当前或未来的决策无关。

Example: $50,000 spent last year to improve a building. This cost should not be included in current project analysis.
示例：去年花费了 50,000 美元用于改善建筑物。这项成本不应包含在当前项目分析中。

Key Insight: Sunk costs do not influence the decision to undertake a new project as they are already incurred and cannot be changed. Only incremental or future costs should be considered.
关键洞察：沉没成本不会影响是否启动新项目的决策，因为它们已经发生且无法改变。只有增量成本或未来成本应被考虑。

Example 2
示例 2

Scenario: $10,000 fee was paid last year for a feasibility study on a project. It costs $125,000 today as an initial investment. Expected NCFs is $75,000 for each year for next 2 years. Discount rate is 10%.
情景：去年支付了 10,000 美元用于一项项目的可行性研究。如今作为初始投资需要 125,000 美元。预计未来两年每年的净现金流量为 75,000 美元。折现率为 10%。

Will you accept this project?
你会接受这个项目吗？

Problem:
问题：

Sunk Cost: $10,000 paid last year for feasibility study (ignore as it's already incurred).
沉没成本：去年为可行性研究支付了 10,000 美元（忽略，因为它已经发生）。

Initial Investment: $125,000 (consider this).
初始投资：125,000 美元（考虑）。

Expected Net Cash Flows (NCF): $75,000 annually for 2 years.
预期净现金流量（NCF）：每年 75,000 美元，共 2 年。

Discount Rate: 10%.
折现率：10%。

Example 3
示例 3

Scenario: Company owns machinery which can be leased at $3m per year, or used in a project which yields $20m pa for next 2 years and costs $30m today (opportunity cost of capital is 10%) 公司拥有的机器设备可以以每年 300 万美元的价格出租，或用于未来 2 年年收益为 2000 万美元、目前成本为 3000 万美元的项目（资本机会成本为 10%）
情景：公司拥有的机器设备可以以每年 300 万美元的价格出租，或用于未来 2 年年收益为 2000 万美元、目前成本为 3000 万美元的项目（资本机会成本为 10%）

Should the project be taken on?
是否应该进行该项目？

Problem:
问题：

The company has machinery that can either:
公司现有的机械设备可以：

Be leased for $3 million per year (opportunity cost).
以每年 300 万美元的租金租赁（机会成本）。

Be used in a project that yields $20 million per year for 2 years but costs $30 million upfront.
用于一个项目，该项目在 2 年内每年产生 2000 万美元的收入，但需要前期投入 3000 万美元。

Opportunity cost of capital: 10%.
资本的机会成本：10%。

Beware of Allocated Overheads
警惕分配的间接费用

Existing Overheads:
现有的间接费用：

Do not include fixed costs that already exist in the analysis of a new project.
不要将已存在于新项目分析中的固定成本包含在内。

Relevant Costs:
相关成本：

Only include changes in fixed costs directly resulting from the proposed project.
仅包括因拟议项目直接引起的固定成本变化。

Examples of Relevant Changes:
相关变化示例：

Increase/decrease in salaries.
工资的增加/减少。

Increase/decrease in rent.
租金的增加/减少。

Increase/decrease in utility expenses (e.g., power).
公用事业费用的增加/减少（例如，电力）。

Key Takeaway: Focus on incremental costs or savings directly tied to the project. Ignore pre-existing fixed overheads.
要点：关注与项目直接相关的增量成本或节约。忽略现有的固定管理费用。

Net Working Capital (NWC)
营运资本 (NWC)

Definition:
定义：

Net Working Capital (NWC) = Current Assets (CA) - Current Liabilities (CL).
净营运资本（NWC）= 流动资产（CA）- 流动负债（CL）。

Key Components:
关键组成部分：

Manufacturing Support:
制造支持：

Extra raw materials and work-in-progress inventory to sustain production.
额外原材料和在制品库存以维持生产。

Selling Support:
销售支持：

Additional finished goods inventory and debtor accounts to aid sales activities.
额外成品库存和应收账款以辅助销售活动。

Key Takeaway: Properly manage and allocate NWC as it directly supports operational and sales processes.
要点：正确管理和分配营运资本，因为它直接支持运营和销售流程。

Financing Costs
融资成本

Interest Costs:
利息成本：

Should not be included as explicit cash flows in project evaluation.
不应作为明确的现金流在项目评估中包含。

Reasons:
原因：

Interest costs are already accounted for in the discount rate or the required rate of return, which represents the company’s cost of capital.
利息成本已在折现率或要求的回报率中考虑，这代表了公司的资本成本。

If the project’s NPV (Net Present Value) is positive, it implies that the cash flows generated will adequately cover interest costs and other financing expenses.
如果项目的净现值（NPV）为正，这意味着产生的现金流将足以覆盖利息成本和其他融资费用。

Key Takeaway: Focus on operating cash flows for project evaluation, as financing is implicitly considered through the discount rate.
要点：在项目评估中应关注经营现金流，因为融资通过折现率隐含考虑在内。

Impact of Inflation
通货膨胀的影响

Inflation's Role:
通货膨胀的作用：

An essential consideration in capital budgeting, as it affects the value of future cash flows.
资本预算中的一个重要考虑因素，因为它会影响未来现金流的价值。

Interest Rate Types:
利率类型：

Nominal Interest Rate: Includes inflation.
名义利率：包含通货膨胀。

Real Interest Rate: Excludes inflation.
实际利率：排除通货膨胀。

Cash Flow Discounting:
现金流量折现：

Nominal Cash Flows: Discounted at the nominal rate.
名义现金流量：按名义利率折现。

Real Cash Flows: Discounted at the real rate.
实际现金流量：按实际利率折现。

Key Takeaway: Always match the type of cash flows (nominal or real) with the corresponding interest rate to ensure accurate evaluation.
要点：始终将现金流量类型（名义或实际）与相应的利率相匹配，以确保准确评估。

Example: NPV Calculation
示例：净现值计算

Scenario:
情景：

Cash Flows (CF):
现金流（CF）：

Year 0: −$1000
第 0 年：−$1000

Year 1: +$600
第 1 年：+$600

Year 2: +$650
第 2 年：+$650

Nominal Discount Rate: 14% per annum.
名义折现率：每年 14%。

Inflation Rate: 5% per annum.
通货膨胀率：每年 5%。

Calculation:
计算：

Interpretation:
解读：

The project has a positive NPV (+$26.47).
该项目的净现值为正（+$26.47）。

Decision: Accept the project since it adds value.
决策：接受该项目，因为它能增加价值。

Consistency in the Treatment of Inflation
通货膨胀处理的连贯性

Scenario:
情景：

Initial amount: $100
初始金额：100 美元

Investment rate: 10% annual return.
投资利率：10%年回报率。

Inflation rate: 6% annual increase in hamburger prices.
通货膨胀率：汉堡价格每年增长 6%。

Calculation:
计算：

Key Insight:
关键洞见：

Real return (3.8%) is lower than nominal return (10%) due to the impact of inflation.
由于通货膨胀的影响，实际回报率（3.8%）低于名义回报率（10%）。

Adjust nominal returns to account for inflation when evaluating purchasing power.
调整名义回报以考虑通货膨胀，从而评估购买力。

Fisher Relation: Formalizing the Effect of Inflation 费雪关系：通货膨胀效应的形式化
费雪关系：通货膨胀效应的形式化

Variables:
变量：

R: Nominal rate of return.
R: 名义回报率。

r: Real rate of return.
r: 实际回报率。

ρ: Inflation rate.
ρ: 通货膨胀率。

Relationship:
关系：

Explanation:
解释：

Nominal Rate of Return (R): Includes both the real return and the effect of inflation.
名义回报率（R）：包括实际回报和通货膨胀的影响。

Real Rate of Return (r): Excludes inflation, reflecting the actual purchasing power growth.
实际收益率（r）：排除通货膨胀，反映实际购买力的增长。

The formula adjusts nominal returns to account for inflation, providing a true reflection of real gains.
该公式调整名义收益率以考虑通货膨胀，提供真实的实际收益反映。

Cash Flow Analysis: Overview
现金流分析：概述

Key Points:
要点：

Three-Step Process:
三步法：

A structured approach is used for analyzing cash flows.
采用结构化方法分析现金流。

Step 1: Calculating Incremental After-Tax Cash Flows - Initial Cash Outflow (ICO)
步骤 1：计算增量税后现金流 - 初始现金流出（ICO）

Key Elements:
关键要素：

Definition:
定义：

Initial cash outflow refers to the net cash investment required at the beginning or setup phase of a project.
初始现金流出是指在项目开始或设置阶段所需的净现金投资。

Components to Consider:
需要考虑的组成部分：

Net Working Capital (NWC): Adjustments for changes in working capital needs.
营运资本（NWC）：调整工作资本需求变化。

Sale of Old Assets: If applicable, include profits and tax effects from selling old assets.
出售旧资产：如果适用，应包括出售旧资产产生的利润和税收影响。

Timing:
时间：

This outflow usually occurs at time t = 0, marking the starting point of the project.
这项支出通常发生在时间 t=0，标志着项目的开始。

Breakdown of Initial Cash Outflow (ICO)
初始现金流出（ICO）的分解

Cost of New Asset(s):
新资产成本：

Include the purchase price of the new asset(s) being acquired.
包括新资产（件）的购买价格。

Capitalized Expenditures:
资本性支出：

Add expenditures related to installation, delivery, or modifications necessary to make the asset operational.
增加与安装、交付或使资产投入运营所需的修改相关的支出。

Changes in Net Working Capital (NWC):
净营运资本（NWC）的变化：

Increase (+): If the project requires additional working capital.
增加（+）：如果项目需要额外营运资金。

Decrease (-): If the project frees up working capital.
减少（-）：如果项目释放了营运资金。

Net Proceeds from Sale of Old Asset(s):
出售旧资产（s）的净收益：

Subtract the proceeds earned from selling any old asset(s) if they are being replaced.
如果旧资产（s）被替换，则减去出售任何旧资产（s）所得的收益。

Tax Adjustments:
税收调整：

Tax paid (+) or savings (-) arising from the sale of old assets should be accounted for. This reflects the tax effect of any gain or loss on the sale.
出售旧资产产生的税收支付（+）或节省（-）应予以核算。这反映了出售中任何收益或损失的税收影响。

Final Calculation:
最终计算：

Combine the above components to derive the Initial Cash Outflow.
将上述各部分结合起来，得出初始现金流出。

Step 2: Calculating Incremental After-tax Cash Flows (NCFs)
步骤 2：计算增量税后现金流量（NCF）

Definition:
定义：

Incremental NCFs refer to the net cash flows generated by a project after the initial cash investment is made but excluding the final period's cash flow.
增量 NCF 指的是在初始现金投资之后由项目产生的净现金流量，但不包括最终时期的现金流量。

Key Aspects:
关键要点：

Incremental:
增量：

Only includes cash flows directly attributable to the project or decision. This excludes unrelated business cash flows.
仅包括直接归属于项目或决策的现金流。这排除了与业务无关的现金流。

After-tax:
税后：

Adjusted for taxes to reflect the actual cash impact on the firm.
经过税收调整，以反映对公司的实际现金流影响。

Excludes Final Period's CF:
排除最终期现金流量：

Focuses on operational cash flows during the project's lifecycle but excludes the final recovery or terminal value.
关注项目生命周期内的经营现金流量，但排除最终回收或终值。

Components of Incremental Net Cash Flows (NCFs)
增量净现金流量（NCFs）的组成部分

Net increase (or decrease) in operating revenue
经营收入的净增加（或减少）

Adjust for any increase or decrease in revenue.
调整任何收入的增加或减少。

Subtract (or add) any net change in operating expenses (excluding depreciation).
减去（或加上）任何营业费用（不包括折旧）的净变化。

Subtract (or add) net increase (or decrease) in tax depreciation charges
减去（或加上）税收折旧费用的净增加（或减少）

Adjust income before taxes by including the effect of tax-deductible depreciation.
通过包含税收抵扣折旧的影响来调整税前收入。

Net change in income before taxes
税前收入的净变化

This represents taxable income after considering revenue, operating expenses, and tax depreciation adjustments.
这表示在考虑收入、运营费用和税收折旧调整后的应税收入。

Subtract (or add) net increase (or decrease) in taxes
减去（或加上）税收的净增加（或减少）

Taxes calculated based on the company’s applicable tax rate.
根据公司适用的税率计算税收。

Net change in income after taxes
税后收入的净变化

The post-tax income after adjustments for revenues, expenses, depreciation, and taxes.
调整收入、费用、折旧和税收后的税后收入。

Add back net increase (or decrease) in tax depreciation charges
加上税后折旧费用的净增加（或减少）

Depreciation is a non-cash expense and is added back to compute cash flows.
折旧是一项非现金费用，需要加回来以计算现金流。

Incremental Net Cash Flow for the period
期间增量净现金流

The final net cash flow for the period after all adjustments and tax effects.
经过所有调整和税收效应后的期间最终净现金流。

Terminal-Year Incremental Net Cash Flows (NCFs) - Step 3
终年增量净现金流量（NCFs）- 第 3 步

Definition: Terminal-year NCFs represent the final period’s incremental after-tax cash flows, marking the last stage of the project after its completion.
定义：终年 NCFs 代表项目完成后的最后时期的增量税后现金流量，标志着项目最后的阶段。

Timing:
时间：

Typically occurs in the final year of the project.
通常发生在项目的最后一年。

May also include additional cash flows extending over several years if the end-of-project stage spans longer.
也可能包括跨越数年的额外现金流，如果项目结束阶段持续时间较长。

Key Considerations:
主要考虑因素：

Salvage Value: The residual or sale price of equipment at the end of its useful life.
残值：设备在其使用寿命结束时的剩余价值或销售价格。

Profit/Loss vs. Book Value (BV):
利润/亏损与账面价值（BV）：

If the asset is sold for more or less than its book value, the resulting profit or loss has tax implications.
如果资产出售价格高于或低于其账面价值，由此产生的利润或损失会产生税务影响。

Repayment of Net Working Capital (NWC):
净营运资本（NWC）的偿还：

Any funds tied up in working capital (e.g., inventory or accounts receivable) are typically recovered at the end.
被投入到营运资本（例如库存或应收账款）的资金通常在最后收回。

Terminal-Year Incremental After-Tax Cash Flows (NCFs) Components
期末增量税后现金流（NCFs）组成部分

Incremental NCF for Terminal Period:
期末增量净现金流量：

Calculate the incremental cash flow generated in the final period of the project.
计算项目最后阶段产生的增量现金流。

Salvage Value (Disposal or Reclamation Costs):
残值（处置或回收成本）：

Include any proceeds from the sale of assets or adjust for reclamation costs incurred.
包括资产出售的任何收益，或调整所发生的回收成本。

Net salvage value = Sale proceeds - Disposal costs
净残值 = 销售收入 - 处置成本

Taxes (or Tax Savings):
税收（或税收节省）：

Adjust for taxes or tax savings due to the disposal of assets or new assets sold.
调整因资产处置或新资产销售而产生的税收或税收节省。

Tax effect = Sale price - Book value (if gain or loss exists)×Tax rate
税收影响 = 销售价格 - 账面价值（如有盈亏）× 税率

Changes in Net Working Capital (NWC):
净营运资本（NWC）的变化：

Adjust for changes in NWC due to project completion:
调整因项目完成导致的 NWC 变化：

Decrease in NWC: Results in cash inflow.
NWC 减少：导致现金流入。

Increase in NWC: Results in cash outflow.
NWC 增加：导致现金流出。

Final Terminal NCF:
最终终年净现金流：

Sum the above components to determine the final terminal-year cash flows.
将上述各部分相加，以确定最终期末年的现金流量。

Comprehensive Example 4
综合示例 4

Current Situation
当前情况

Old Machine:
旧机器：

Original Cost: $20,000
原始成本：$20,000

Life: 10 years; currently in use for 5 years.
使用寿命：10 年；目前已使用 5 年。

Depreciation: Straight-line; value reduced to $0 over 10 years.
折旧：直线折旧；10 年内价值减少至$0。

Salvage Value: $15,000.
残值：$15,000。

Operational Costs: Salary $15,000/year, Fringe Benefits $2,000/year.
运营成本：工资 15,000 美元/年，津贴 2,000 美元/年。

Maintenance: $7,000/year.
维护：每年 7,000 美元。

Defect Costs: $3,000/year.
缺陷成本：3,000 美元/年。

Proposed Investment
拟议投资

New Machine:
新机器：

Purchase Cost: $50,000.
购置成本：50,000 美元。

Salvage Value: $10,000 after 5 years.
残值：5 年后为 10,000 美元。

Depreciation: Straight-line over 5 years.
折旧：5 年内直线折旧。

Additional Costs to Operate:
运营额外成本：

Shipping: $3,000.
运输：3,000 美元。

Installation: $2,000.
安装：2,000 美元。

Inventory Increase: $5,000.
库存增加：5,000 美元。

Operational Costs:
运营成本：

Maintenance: $2,000/year.
维护费用：每年 2,000 美元。

Defect Costs: $4,000/year.
缺陷成本：每年 4,000 美元。

Training:
培训：

A similar $5,000 training program occurred recently.
最近发生了一个类似的 5000 美元培训项目。

Uncertainty whether to charge $2,500 (half) to the project.
不确定是否应该将 2500 美元（一半）记入该项目。

Financial Details
财务详情

Tax Rate: 47%.
税率：47%。

Financing: Borrow $20,000 at 10% interest. Additional payments of $2,000/year.
融资：借入 20,000 美元，利率 10%。每年额外支付 2,000 美元。

Required Rate of Return: 20%.
要求回报率：20%。

Required:
要求：

What is the project’s initial investment?
这个项目的初始投资是多少？

What are the incremental cash flows over the project’s life?
这个项目在其生命周期中的增量现金流是多少？

What is the terminal cash flow?
期末现金流是多少？

What is the NPV? IRR?
净现值（NPV）是多少？内部收益率（IRR）是多少？

Should the project be accepted (yes/no)? Why/why not?
这个项目应该被接受吗（是/否）？为什么/为什么不？

Incremental Analysis (Initial Investment)
增量分析（初始投资）

Initial Cash Outflow (Time = 0):
初始现金流出（时间=0）：

New Unit Cost: $55,000 (outflow).
新单位成本：55,000 美元（流出）。

Old Unit Sale Proceeds: $15,000 (inflow).
旧设备出售所得：15,000 美元（流入）。

Tax on Sale of Old Unit: $2,350 (outflow).
出售旧设备的税：$2,350（流出）。

Tax calculated based on the taxable gain: Sale Price ($15,000) - Book Value ($10,000).
基于应税收益计算的税：出售价格（$15,000）- 账面价值（$10,000）。

Taxable Gain = $5,000; Tax = $5,000 × 47% = $2,350.
应税收益 = $5,000；税额 = $5,000 × 47% = $2,350。

Increase in Net Working Capital (NWC): $5,000 (outflow).
净营运资本（NWC）的增加：$5,000（流出）。

Net Initial Investment:
净初始投资：

Total = New Unit Cost - Old Unit Sale Proceeds + Tax + Increase in NWC.
总计 = 新单位成本 - 旧单位销售所得 + 税 + NWC 的增加。

Net Outflow = $55,000 - $15,000 + $2,350 + $5,000 = $47,350.
净流出 = $55,000 - $15,000 + $2,350 + $5,000 = $47,350.

Incremental Analysis (Operating Cash Flows)
增量分析（经营现金流）

Incremental Revenue (New - Old):
增量收入（新 - 旧）：

$0 for all periods, as no change in revenue is stated.
所有期间均为 0，因为没有说明收入发生变化。

Salary Savings:
工资节省：

$17,000 for all periods (new machine eliminates old operator's salary and benefits).
所有期间均为 $17,000（新机器消除了旧操作员的工资和福利）。

Incremental Costs (New - Old):
增量成本（新 - 旧）：

$4,000 per year due to increased costs of maintenance and defects.
每年增加维护成本和缺陷导致支出$4,000。

Depreciation:
折旧：

New machine depreciation: $(11,000) per year.
新机器折旧：每年$(11,000)。

Add back old machine depreciation: $2,000 per year.
加回旧机器折旧：每年$2,000。

EBIT (Earnings Before Interest and Tax):
EBIT（税前利润）：

$12,000 for all periods, calculated as:
所有期间均为 12,000 美元，计算方法如下：

$17,000 (Salary Savings) - $4,000 (Incremental Costs) - $11,000 (Depreciation New) + $2,000 (Depreciation Old).
17,000 美元（工资节省）- 4,000 美元（增量成本）- 11,000 美元（新增折旧）+ 2,000 美元（旧折旧）。

Tax:
税收：

$(5,640) per year, calculated at a 47% tax rate on EBIT.
每年 5,640 美元，按 EBIT 的 47%税率计算。

EAT (Earnings After Tax):
EAT（税后收益）：

$6,360 per year, calculated as:
每年 6,360 美元，计算方法如下：

$12,000 (EBIT) - $5,640 (Tax).
12,000 美元（EBIT）- 5,640 美元（税）。

Add Back Depreciation:
加回折旧：

$9,000 per year ($11,000 depreciation for new machine - $2,000 depreciation for old machine).
每年 9,000 美元（新机器折旧 11,000 美元 - 旧机器折旧 2,000 美元）。

Operating Cash Flow:
经营现金流：

$15,360 per year for all periods, calculated as:
每年 15,360 美元，计算如下：

$6,360 (EAT) + $9,000 (Depreciation).
6,360 美元（税后利润）+ 9,000 美元（折旧）。

Terminal Cash Flows (Ex 4 - Incremental Analysis)
终期现金流量（例 4 - 增量分析）

Components of Terminal Cash Flows (T = 5):
终期现金流量的组成部分（T = 5）：

Salvage Value:
残值：

Value recovered from the sale of the new asset at the end of its useful life: $10,000.
新资产在其使用寿命结束时通过出售收回的价值：$10,000。

Tax Paid on Capital Gain:
资本利得所缴纳的税额：

Tax calculated on the difference between salvage value and book value (if a gain is realized): $(4,700).
在残值与账面价值之间的差额（如果实现收益）计算的税额：$(4,700)。

Recovery of NWC (Net Working Capital):
营运资本（NWC）的收回：

The previously increased NWC is recovered at the end of the project: $5,000.
项目结束时收回之前增加的营运资本：$5,000。

Final Year Operating Cash Flow:
最终年经营现金流：

Cash flow generated from regular operations in the final year: $15,360.
最终年产生的常规运营现金流：$15,360。

Net Terminal Cash Flow Calculation:
净终值现金流计算：

Salvage Value: $10,000
残值：$10,000

Less: Tax on Capital Gain: $(4,700)
减：资本利得税：$(4,700)

Add: Recovery of NWC: $5,000
加：回收营运资本：$5,000

Add: Final Year Operating Cash Flow: $15,360
加：最终年经营现金流：$15,360

Net Terminal Cash Flow = $25,660
净终结现金流 = $25,660

NPV Calculation (Ex 4 - Incremental Analysis)
NPV 计算（例 4 - 增量分析）

Net Present Value (NPV):
净现值（NPV）：

Initial Investment (C₀): $(-47,350) (negative cash flow at time = 0)
初始投资（C₀）：$(-47,350)（时间=0 时的负现金流）

Operating Cash Flows (C₁ to C₄): $15,360 annually for 4 years
经营现金流（C₁至 C₄）：每年$15,360，共 4 年

Terminal Cash Flow (C₅): $25,660 at year 5
终期现金流（C₅）：第 5 年$25,660

Discount Rate (i):
折现率（i）：

Required Rate of Return: 20%
所需回报率：20%

Calculation Using Sharp EL-735 Financial Calculator:
使用 Sharp EL-735 财务计算器计算：

Interpretation:
解读：

Since the NPV is positive ($2,725.14), the project is expected to generate value above the required rate of return and is financially viable. It should be accepted.
由于净现值（NPV）为正（$2,725.14），该项目预计将产生高于所需回报率的收益，并且具有财务可行性。应接受该项目。

Compute IRR:
计算内部收益率（IRR）：

IRR = 22.41%

Interpretation:
解读：

Since the IRR (22.41%) is greater than the required rate of return (20%), the project is financially viable and should be accepted. IRR indicates that the project will generate returns higher than the cost of capital.
由于内部收益率（22.41%）高于要求的回报率（20%），该项目在财务上是可行的，应该被接受。内部收益率表明该项目将产生的回报高于资本成本。

Incremental Analysis (Decision)
增量分析（决策）

Conclusion:
结论：

The project should be accepted as it generates wealth for shareholders based on the following criteria:
根据以下标准，该项目应该被接受，因为它为股东创造了财富：

NPV > 0: The Net Present Value (NPV) of the project is positive, indicating it adds value.
NPV > 0: 项目的净现值（NPV）为正，表明其能增加价值。

IRR > Cost of Capital: The Internal Rate of Return (IRR) exceeds the required rate of return, proving the project's profitability.
IRR > 成本资本：内部收益率（IRR）超过要求的回报率，证明项目的盈利能力。

Projects with unequal lives
寿命不等的投资项目

Example 5 - Mutually Exclusive Projects with Different Lives
示例 5 - 寿命不同的互斥项目

Problem:
问题：

A factory needs to choose between two air cleaners with different costs and lifespans:
一个工厂需要在两种不同成本和寿命的空气净化器之间做出选择：

Cleaner X:

Initial Cost: $4,000
初始成本：$4,000

Annual Operating Cost: $100
年运营成本：100 美元

Lifespan: 10 years
使用寿命：10 年

Cleaner Y:
清洁工 Y：

Initial Cost: $1,000
初始成本：$1,000

Annual Operating Cost: $500
年运营成本：$500

Lifespan: 5 years
使用寿命：5 年

Decision:
决策：

The cost of capital for this investment is 10%.
这项投资的资本成本为 10%。

The choice depends on calculating the Equivalent Annual Cost (EAC) for each option to determine the more cost-effective cleaner over their respective lifespans.
选择取决于计算每个选项的等年成本（EAC），以确定在其各自的使用寿命内更具成本效益的清洁设备。

NPV Analysis:
净现值分析：

Cleaner X:

Initial Cost: -$4,000
初始成本：-4,000

Present Value of Operating Costs: ∑ (100 / 1.1ᵗ) for t = 1 to 10
经营成本现值：∑ (100 / 1.1ᵗ) 对于 t = 1 到 10

NPVₓ ≈ -$4,614.46
NPVₓ ≈ -4,614.46

Cleaner Y:
清洁工 Y：

Initial Cost: -$1,000
初始成本：-1,000

Present Value of Operating Costs: ∑ (500 / 1.1ᵗ) for t = 1 to 5
经营成本现值：∑ (500 / 1.1ᵗ) 对于 t = 1 到 5

NPVᵧ ≈ -$2,895.39
NPVᵧ ≈ -2,895.39

Observation:
观察：

At first glance, Cleaner Y has a lower NPV (less negative) than Cleaner X.
%% 一瞥之下，清洁器 Y 的净现值（负值较小）低于清洁器 X。

Key Insights:
关键洞察：

Cleaner X lasts twice as long (10 years) as Cleaner Y (5 years).
清洁器 X 的使用寿命是清洁器 Y 的两倍（10 年），而清洁器 Y 为 5 年。

While Cleaner Y appears to have a lower NPV initially, the longer lifespan of Cleaner X reduces its effective cost over time.
虽然清洁器 Y 的初始净现值较低，但清洁器 X 更长的使用寿命使其长期有效成本降低。

Investments of Unequal Lives
使用寿命不等的投资

When comparing projects with different lifespans, two main methods are used:
在比较具有不同生命周期的项目时，通常使用两种主要方法：

Matching Cycle or Lowest Common Multiple (LCM) Method
匹配周期或最小公倍数（LCM）方法

Repeat the projects until their lifespans align to make direct comparisons.
重复这些项目，直到它们的寿命期一致，以便进行直接比较。

Equivalent Annual Annuity (EAA) or Equivalent Annual Cost (EAC) Method
等额年金法（EAA）或等额成本法（EAC）

Converts the Net Present Value (NPV) of each project into an equivalent annual amount for better comparability.
将每个项目的净现值（NPV）转换为等额年金额，以便更好地进行比较。

Key Insight:
关键洞见：

Use EAC or EAA when the projects have unequal durations and repeated cycles are impractical.
当项目持续时间不等且重复周期不实用时，使用 EAC 或 EAA。

Matching Cycle or Lowest Common Multiple Method
匹配周期或最小公倍数法

The cash flow timelines for Cleaner X and Cleaner Y are extended to a common duration (10 years) to enable a fair comparison. Here's the breakdown:
将 Cleaner X 和 Cleaner Y 的现金流量时间线扩展到相同持续时间（10 年），以进行公平比较。具体如下：

NPV Comparison for Mutually Exclusive Projects with Different Lives
不同寿命互斥项目的 NPV 比较

The Net Present Value (NPV) for Cleaner X and Cleaner Y is recalculated to account for their differing project lives.
净现值（NPV）对清洁器 X 和清洁器 Y 进行了重新计算，以考虑它们不同的项目寿命。

Cleaner X:

NPV ≈ -$4614.46

Cleaner Y:
清洁工 Y：

NPV ≈ -$4693.20

Key Takeaway:
要点：

When comparing on an equal time frame, Cleaner X proves to be cheaper (less negative NPV). This indicates Cleaner X is the more cost-effective choice over a 10-year duration.
在相同的时间框架内进行比较时，清洁剂 X 被证明更便宜（NPV 更低）。这表明在 10 年的使用周期内，清洁剂 X 是更具成本效益的选择。

Equivalent Annual Annuity (EAA) Comparison
等额年金比较 (EAA)

The Equivalent Annual Cost (EAC) is calculated for both Cleaner X and Cleaner Y to compare their costs over unequal lifespans using the Present Value of Annuity Factor (PVAF).
等年成本（EAC）使用年金现值系数（PVAF）计算 Cleaner X 和 Cleaner Y，以比较它们在寿命不等情况下的成本。

Cleaner X:

NPVₓ,₀ = -$4000 - Σ [100/(1.1)^t]

EACₓ ≈ $750.98

Cleaner Y:
清洁工 Y：

NPVᵧ,₀ = -$1000 - Σ [500/(1.1)^t]

EACᵧ ≈ $763.80

Key Takeaway:
关键要点：

Cleaner X is cheaper based on the EAC calculation, making it the better option despite differing lifespans.
基于 EAC 计算，清洁器 X 更便宜，尽管使用寿命不同，因此它是更好的选择。

Key Considerations for Using the Equivalent Annual Annuity (EAA) Method
使用等额年金（EAA）方法的关键考虑因素

The EAA method assumes projects are repeated indefinitely.
EAA 方法假设项目无限期重复。

Use the method only when:
仅在以下情况下使用该方法：

There is a high probability of replicating projects in the future.
未来复制项目的高概率。

Projects have significantly different lives.
项目寿命差异显著。

Unequal life analysis is irrelevant for:
不等寿命分析对以下情况无关紧要：

Independent projects.
独立项目。

It is only applicable to mutually exclusive projects.
仅适用于互斥项目。

Example 6
例 6

How do we choose between the following two projects (assume the cost of capital is 11.5% pa)
我们如何在这两个项目之间进行选择（假设资本成本为每年 11.5%）

Example 6 (Continued)
示例 6（续）

NPV Comparison:
净现值比较：

NPV A >NPV B, so Project A seems better at first glance.
NPV A > NPV B，所以项目 A 初看似乎更优。

Key Considerations:
关键考虑因素：

Project B can be reinvested in 3 years, potentially increasing shareholder value (not included in current calculations).
项目 B 可以在 3 年内再投资，有可能增加股东价值（未包含在当前计算中）。

Repeating Project B could align its timeline with Project A, making a fair comparison of NPVs possible over the same time horizon.
重复项目 B 可以使它的时间线与项目 A 一致，从而在同一时间范围内进行 NPV 的公平比较。

Conclusion:
结论：

While NPV A >NPV B initially, the reinvestment opportunity for Project B needs to be considered for a comprehensive analysis.
虽然初始时 NPV A > NPV B，但需要考虑项目 B 的再投资机会进行全面分析。

Example 6 (Continued - Method 1: Replacement Chain)
示例 6（续 - 方法 1：替换链）

Replacement Chain Method:
替换链方法：

Aligns the timelines of projects by replicating Project B until its life equals Project A’s.
通过复制项目 B 直到其寿命等于项目 A 的寿命来对齐项目的时序线。

Process:
流程：

Compute the NPV for Project B over its first 3-year cycle (${''NPV''}_{\mathrm{}}$).
计算项目 B 在其第一个 3 年周期（ ${''NPV''}_{\mathrm{}}$ ）的净现值。

Value: $5391.
价值：$5391。

Present value at 11.5%: $3889.
11.5%的现值：$3889。

Replicate Project B for another 3 years (second cycle).
将项目 B 再复制 3 年（第二个周期）。

Extend NPV of Project B to cover 6 years.
将项目 B 的净现值扩展到 6 年。

Total extended NPV over both cycles: $9280.
总扩展净现值（两个周期）：$9280。

Key Insight:
关键洞见：

This method enables a fair comparison of NPV by equating the project timelines.
这种方法通过使项目时间线相等，实现了对 NPV 的公平比较。

Example 6 (Continued - Replacement Chain)
例 6（继续 - 替换链）

Project B's NPV for the first 3 years is $5,391.
项目 B 在前 3 年的 NPV 为 5,391 美元。

NPV of repeated projects (second cycle) is also $5,391.
重复项目的 NPV（第二周期）也为 5,391 美元。

Discount the second 3-year cycle's NPV (${''NPV''}_{\mathrm{}}$) to time t=0:
将第二个 3 年周期的净现值（ ${''NPV''}_{\mathrm{}}$ ）折算到 t=0 时刻：

Comparison:
比较：

NPV B is greater than NPV A ($7,686).
NPV B 大于 NPV A（$7,686）。

Choose Project B, as it generates more value when repeated.
选择项目 B，因为它在重复时能创造更多价值。

Method 2 - Equivalent Annual Annuity (EAA) Approach
方法 2 - 等额年金法（EAA）

Definition:
定义：

The EAA method (also known as Annual Equivalent Cost - AEC or Equivalent Annual Value - EAV) is used to compare projects of different durations.
等额年金法（也称为年等值成本法 - AEC 或年等值价值法 - EAV）用于比较不同持续期的项目。

Steps:
步骤：

Calculate the NPV:
计算 NPV：

Determine each project's Net Present Value (NPV) over its initial life.
确定每个项目的净现值（NPV）在其初始生命周期内。

Find the Equivalent Annual Annuity:
计算等额年金：

For each project, calculate the constant annual cash flow (annuity) over the project's life that results in the same NPV.
对于每个项目，计算在其生命周期内产生相同 NPV 的恒定年度现金流（年金）。

Step 2 (Detailed):
步骤 2（详细）：

Calculate the Equivalent Annual Annuity (EAA):
计算等额年金（EAA）：

Find the constant annual cash flow (denoted as C) of an ordinary annuity over the project’s life that equals the NPV.
找出在项目生命周期内等于净现值的普通年金的恒定年现金流（用 C 表示）。

Step 3:
步骤 3：

Compare the EAA values:
比较 EAA 值：

Profitable Projects: Choose the project with the largest EAA value.
盈利项目：选择 EAA 值最大的项目。

Costly Projects: Choose the project with the lowest EAA value, if the cost is unavoidable.
成本项目：如果成本不可避免，选择 EAA 值最小的项目。

Example 6 Again - EAA Calculation
例 6 再次 - EAA 计算

Explanation:
解释：

For Project B, the Equivalent Annual Annuity (EAA) is calculated by distributing the project's NPV over its duration as an annuity.
对于项目 B，等额年金（EAA）是通过将项目的净现值（NPV）在其持续时间上按年金形式分配来计算的。

Task:
任务：

Solve for C (EAA) using the PVIFA formula:
使用 PVIFA 公式求解 C（EAA）：

Result
结果

Conclusion:
结论：

Since EAA B >EAA A, Project B is preferred when considering equivalent annual cash flows.
由于 EAA B > EAA A，在考虑等额年现金流量时，项目 B 更受青睐。

Example 6 - Final Decision
示例 6 - 最终决策

Key Results:
关键结果：

EAA B = $2,225.28 per year (for 3 years).
EAA B = 每年 $2,225.28（持续 3 年）。

EAA A = $1,843.04 per year (for 6 years).
EAA A = 每年 1,843.04 美元（持续 6 年）。

Conclusion:
结论：

Project B is preferred as EAA B > EAA A.
项目 B 更受青睐，因为 EAA B > EAA A。

Even though Project B has a shorter duration, its annual equivalent annuity provides higher value per year compared to Project A.
尽管项目 B 的持续时间较短，但与项目 A 相比，其年等额年金每年提供的价值更高。

EAA Method: Word of Warning
EAA 方法：注意事项

The Equivalent Annual Annuity (EAA) method assumes projects will be repeated indefinitely.
等额年金（EAA）方法假设项目将无限期重复。

Use the EAA method only if:
仅当：

There is a reasonable probability of replicating projects.
复制项目有合理概率时，才使用 EAA 方法。

Projects have significantly different lives.
项目寿命差异显著。

Note: Unequal life analysis applies only to mutually exclusive projects, not independent ones. 不平等寿命分析仅适用于相互排斥的项目，而不适用于独立的项目
注意：不平等寿命分析仅适用于相互排斥的项目，而不适用于独立的项目。