📖 不均匀现金流 PV 计算

📌 Topic: When PMT Is No Longer Equal — Every Cash Flow Must Be Discounted Individually

1. From Uniform to Uneven: Goodbye to the "One-Formula" World

L091–L093 dealt with cash flows sharing one trait: every period's amount is identical.

But the real world rarely gives you "uniform" cash flows:

• A project: invest $1M in Year 1, earn $200K in Year 2, $450K in Year 3, $800K in Year 4 — no fixed PMT
• A stock: forecasted dividends of $1.20, $1.50, $2.00, $2.60… — not uniform growth
• Your salary: different raises each year — not a constant growth rate

🔑 Uneven Cash Flow: Each period's amount is independent. No fixed pattern. Must discount every single cash flow individually.

2. Core Principle: Break into N Single-CF PVs, Then Sum

2.1 Back to the Source (L089)

L089 taught us: PV of a single cash flow = FV / (1 + r)^n.

For N periods of uneven cash flows? — Break into N single cash flows, discount each one, then sum them all.

$$\boxed{PV = \frac{CF_0}{(1+r)^0} + \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \cdots + \frac{CF_N}{(1+r)^N}}$$

Usually CF₀ occurs today (t=0), and (1+r)⁰ = 1, so CF₀ is not discounted:

$$\boxed{PV = CF_0 + \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \cdots + \frac{CF_N}{(1+r)^N}}$$

2.2 Step-by-Step Example

Scenario: A project generates the following cash flows (t=0 is the initial investment), discount rate 10%. Find total PV.

Step-by-step calculation:

$$PV = -1{,}000 + \frac{300}{1.10} + \frac{400}{1.10^2} + \frac{500}{1.10^3} + \frac{600}{1.10^4} = +388.78$$

💡 This PV = +388.78 is precisely the project's Net Present Value (NPV): discount all future benefits back to today, then subtract the initial investment. What remains is the "excess value."

3. NPV: The Ultimate Judge of Uneven Cash Flows

3.1 Definition

$$\boxed{NPV = \sum_{t=0}^{N} \frac{CF_t}{(1+r)^t}}$$

$$\boxed{NPV = CF_0 + \frac{CF_1}{1+r} + \frac{CF_2}{(1+r)^2} + \cdots + \frac{CF_N}{(1+r)^N}}$$

where r = required rate of return / discount rate / cost of capital.

3.2 NPV Decision Rule (Core CFA Test Point)

Intuition: If you demand a 10% return and NPV = +50, it means the project gives you 10% AND an extra $50 in value on top.

3.3 Deeper Meaning of NPV > 0

NPV = +388.78 means: 1. The project recovers all initial investment (−1,000) 2. It earns the 10% required return every year 3. Beyond that 10%, it creates an additional $388.78 of "excess value"

💡 NPV > 0 = you earn more than you asked for. NPV < 0 = you don't even meet your own requirements.

4. 🔴 TI BA II Plus: The CF Worksheet (Most Important Section)

4.1 Why Annuity Keys Are Not Enough

Annuity keys (N, I/Y, PV, PMT, FV) assume fixed PMT. Uneven cash flows require inputting each period individually — this needs the "CF Worksheet."

4.2 CF Worksheet Structure

🔑 F (Frequency) is a key-saving design: if 3 consecutive years all have +500, just enter C=500, F=3.

4.3 Complete Step-by-Step (Using the Example Above)

Problem: CF₀ = −1,000, CF₁ = 300, CF₂ = 400, CF₃ = 500, CF₄ = 600, r = 10%. Find NPV.

[CF] [2ND] [CLR WORK]      ← Clear CF worksheet

CF0 = -1000  [ENTER] [↓]
C01 = 300    [ENTER] [↓]
F01 = 1      [ENTER] [↓]    ← Default is 1, press [↓] to skip
C02 = 400    [ENTER] [↓]
F02 = 1      [ENTER] [↓]
C03 = 500    [ENTER] [↓]
F03 = 1      [ENTER] [↓]
C04 = 600    [ENTER] [↓]
F04 = 1      [ENTER]

[NPV]                         ← Enter NPV function
I = 10        [ENTER] [↓]
NPV =         [CPT]           → Displays 388.78 ✅

4.4 Using F (Frequency) for Efficiency

Scenario: CF₀ = −5,000, Years 1-5 each +1,200, Year 6 +3,000. r=8%.

[CF] [2ND] [CLR WORK]

CF0 = -5000  [ENTER] [↓]
C01 = 1200   [ENTER] [↓]
F01 = 5      [ENTER] [↓]     ← 5 consecutive years!
C02 = 3000   [ENTER] [↓]
F02 = 1      [ENTER]

[NPV] I=8 [↓] [CPT] → NPV = ?

Manual verification:

Years 1-5: Ordinary Annuity $$PV_{1-5} = 1{,}200 \times PVIFA(8\%, 5) = 1{,}200 \times 3.9927 = 4{,}791.27$$

Year 6: Single CF $$PV_6 = 3{,}000 / 1.08^6 = 3{,}000 \times 0.6302 = 1{,}890.50$$

$$NPV = -5{,}000 + 4{,}791.27 + 1{,}890.50 = +1{,}681.77$$

💡 The essence of the F key: "bundle" regular segments to reduce keystrokes. No need to enter each repeated year individually.

5. IRR: The Discount Rate That Makes NPV = 0

5.1 Definition

$$\boxed{NPV = 0 = \sum_{t=0}^{N} \frac{CF_t}{(1+IRR)^t}}$$

🔑 IRR (Internal Rate of Return): The discount rate that makes NPV exactly equal to 0.

5.2 IRR Decision Rule

NPV and IRR relationship:

💡 In the vast majority of "conventional" projects, NPV and IRR give consistent decisions. Cases where they conflict (mutually exclusive projects + differing cash flow patterns) are covered in later topics.

5.3 TI BA II Plus IRR Calculation

Using the same example (CF₀ = −1,000, CF₁ = 300, CF₂ = 400, CF₃ = 500, CF₄ = 600):

After entering all CFs:
[IRR] [CPT] → Displays IRR = 23.92% (approx)

Verification: when r = 23.92%, NPV ≈ 0.

5.4 IRR Intuition Check

NPV = +388.78 @ 10% ➡ IRR must be > 10% (lower discount rate → higher NPV; higher discount rate → lower NPV — IRR is the point where NPV hits 0).

6. Uneven Cash Flows vs. Annuity: Tool Switching

6.1 When to Use Annuity Keys vs. CF Keys

🔑 Core principle: Is PMT truly constant? Yes → Annuity keys. No → CF worksheet.

6.2 Annuity Verification Method

If a segment of your cash flows is consecutive and identical → annuity formula can verify CF worksheet results.

7. Common Mistakes and Traps

Trap 1: CF₀ Sign Errors 🔴

🔴 CF₀ in the NPV formula is not discounted (Period 0). Do NOT divide CF₀ by (1+r)!

Trap 2: Confusing "N" in NPV 🔴

Annuity keys: N = total number of periods. CF worksheet: there is no separate N key — N is automatically determined by the number of cash flows you enter.

Trap 3: Forgetting to Clear the CF Worksheet 🔴

If you used CF keys previously, residual data will contaminate your current calculation.

Must do before every new problem: [CF] [2ND] [CLR WORK]

Trap 4: F (Frequency) Input Errors 🔴

💡 The F key is where mistakes happen most: after entering C, press [↓] to reach F — always verify F's value before pressing [↓] to the next group.

Trap 5: IRR May Have No Solution / Multiple Solutions

When cash flows change sign more than once (e.g., −100, +200, −50, +300), there may be multiple IRRs.

This is a CFA Level II topic; for Level I, just know the phenomenon exists.

8. Practice Problems

📝 Example 1: Basic NPV

A project requires an initial investment of $10,000 and is expected to generate net cash flows of $4,000, $5,000, and $3,000 over the next 3 years. Discount rate = 10%. The NPV is closest to:

A. −$180 B. $0 C. $180 D. $320

Solution:

NPV = −10,000 + 3,636.36 + 4,132.23 + 2,253.94 = +22.53

Closest to $0 → Answer B

📝 Example 2: CF Worksheet Verification

Use TI BA II Plus to verify Example 1's NPV:

[CF] [2ND] [CLR WORK]
CF0 = -10000 [ENTER] [↓]
C01 = 4000  [ENTER] [↓]  F01=1 [↓]
C02 = 5000  [ENTER] [↓]  F02=1 [↓]
C03 = 3000  [ENTER] [↓]  F03=1 [ENTER]
[NPV] I=10 [ENTER] [↓] [CPT] → 22.53

📝 Example 3: NPV Decision Rule

Two mutually exclusive projects; can only pick one. Discount rate = 12%.

Which project should be chosen? Why?

Solution: Choose A, because NPV_A (+9,264) > NPV_B (+7,583).

🔑 NPV is an absolute measure (dollar amount). For mutually exclusive projects, always pick the one with the higher NPV, not the higher IRR.

📝 Example 4: IRR Calculation

A project has CF₀ = −$20,000, CF₁ = $8,000, CF₂ = $9,000, CF₃ = $10,000. The IRR is closest to:

A. 10% B. 15% C. 17% D. 20%

Solution: Enter into calculator CF worksheet, then [IRR] [CPT] → ~16.9%, closest to 17% → Answer C

Check: if required return r=12%, then IRR (16.9%) > r → ✅ Accept the project.

📝 Example 5: Mixed Cash Flows Using F Key

An investment: initial outlay $15,000, Years 1-4 receive $4,000/year, Years 5-6 receive $6,000/year. Discount rate 9%. NPV is closest to:

A. $1,200 B. $2,300 C. $3,500 D. $4,800

Solution:

Method 1 (group by annuity): Years 1-4: PMT=4,000, n=4, r=9% $$PV_{1-4} = 4{,}000 \times PVIFA(9\%, 4) = 4{,}000 \times 3.2397 = 12{,}958.85$$

Years 5-6: 2-year annuity of PMT=6,000, first discount to t=4 $$PV_{t=4} = 6{,}000 \times PVIFA(9\%, 2) = 6{,}000 \times 1.7591 = 10{,}554.66$$ Discount back to t=0: $$PV_{5-6} = 10{,}554.66 / 1.09^4 = 10{,}554.66 / 1.4116 = 7{,}476.77$$

$$NPV = -15{,}000 + 12{,}958.85 + 7{,}476.77 = +5{,}435.62

Method 3 (CF worksheet):

CF0 = -15000
C01 = 4000, F01 = 4
C02 = 6000, F02 = 2
[NPV] I=9 [CPT] → 5,435.62

💡 F key: F01=4 means 4 consecutive periods of $4,000; F02=2 means 2 consecutive periods of $6,000. Total: 1 (CF₀) + 4 + 2 = 7 cash flows.

📝 Example 6: NPV Sign Logic

Which of the following projects must have a positive NPV?

A. Initial investment of $5,000; receive $1,200/year for 5 years B. Initial investment of $8,000; receive $3,100/year for 3 years C. No initial investment (CF₀=0); receive any positive future cash flows D. None of the above can be guaranteed

Solution: - A: Depends on r. If r is high enough, NPV could be negative → not guaranteed - B: Same, depends on r → not guaranteed - C: CF₀=0 + all future CFs positive → NPV > 0 for ANY positive r! ∑(positive/positive) > 0 ✅ → Answer C

💡 This is a classic CFA "logic-based" NPV question: no numbers given — judge NPV sign from cash flow characteristics alone.

📝 Example 7: NPV and Discount Rate Relationship

For a project with CF₀ = −1,000, CF₁ = +500, CF₂ = +400, CF₃ = +300:

Which statement is correct?

A. Higher discount rate → higher NPV B. Lower discount rate → lower NPV C. When discount rate = IRR, NPV = 1,000 D. When discount rate = IRR, NPV = 0

Solution: Answer D.

🔑 NPV vs. discount rate: r ↑ → NPV ↓ (negative correlation). IRR is, by definition, the r at which NPV = 0.

9. Key Formula Summary

10. Practice Questions

Q1 (Basic NPV)

A project has CF₀ = −$50,000, CF₁ = $18,000, CF₂ = $22,000, CF₃ = $25,000. Discount rate = 10%. NPV is closest to:

A. −$500 B. $2,500 C. $5,000 D. $7,500

Q2 (NPV Decision Rule)

Two mutually exclusive projects, discount rate = 8%: - Project X: CF₀ = −100,000, receive +35,000/year for 4 years - Project Y: CF₀ = −100,000, CF₁=20,000, CF₂=30,000, CF₃=40,000, CF₄=60,000

Given NPV_X = $15,924, NPV_Y = $17,524. Which should be chosen?

A. Project X (higher IRR) B. Project Y (higher NPV) C. Neither D. Both

Q3 (IRR Calculation)

A project has CF₀ = −$10,000, CF₁ = $4,000, CF₂ = $4,000, CF₃ = $5,000. IRR is closest to:

A. 12% B. 14% C. 16% D. 18%

Q4 (CF Worksheet Operations)

Using TI BA II Plus CF worksheet, enter: CF₀=−2,000, C01=600, F01=3, C02=800, F02=2. How many total periods of cash flows are there (including CF₀, counting F repetitions)?

A. 5 periods B. 6 periods C. 3 cash flow groups (C00 + C01 + C02) D. Cannot be determined

Q5 (Logic Question)

A project: CF₀ = −$P (P>0), all future cash flows are positive, discount rate r > 0. Which condition best guarantees NPV > 0?

A. Sum of all future cash flows > P B. Year 1 cash flow > P C. IRR > 0 D. IRR > r

Q6 (CF₀ Trap)

An investor receives a $5,000 signing bonus today, plus $2,000 at the end of each of the next 3 years as salary, discount rate = 6%. The PV of this cash flow stream is closest to:

A. $5,000 + $2,000 × PVIFA(6%, 3) B. $2,000 × PVIFA(6%, 3) C. $5,000 × (1.06)³ + $2,000 × PVIFA(6%, 3) D. $7,000 × PVIFA(6%, 3)

11. Food for Thought

L089–L093 taught you "regular cash flows" — same amount, repeated, one formula does it all.

L094 tells you the truth — real-world cash flows are never regular. Every single one must be discounted individually. Each one is unique.

That's why the CF worksheet is one of the most important hidden features on the TI BA II Plus: it automates the tedious process of "discount each CF + sum them up."

Once you understand NPV, you've mastered the single most important decision-making tool in corporate finance — no serious company does capital budgeting without NPV.

Once you understand IRR, you can answer that classic question: "What's this project's rate of return, really?"

Starting from L095, we complete the final piece of the TVM puzzle: ways of quoting interest rates — nominal rate, effective annual rate, continuous compounding. Same underlying annual rate, different quoting convention, dramatically different results.

📎 Answers

Explanations:

• Q1: CF₀=−50,000, CF₁=18,000, CF₂=22,000, CF₃=25,000, r=10% PV₁=18,000/1.10=16,363.64 | PV₂=22,000/1.10²=18,181.82 | PV₃=25,000/1.10³=18,782.87 NPV = −50,000 + 16,363.64 + 18,181.82 + 18,782.87 = +3,328.33 → Closest to B ($2,500)

• Q2: Mutually exclusive projects → pick higher NPV. NPV_Y ($17,524) > NPV_X ($15,924) → B ✅

• Q3: CF₀=−10,000, CF₁=4,000, CF₂=4,000, CF₃=5,000. Calculator IRR CPT → ~14.03% → B ✅

• Q4: F01=3 means 3 periods of $600, F02=2 means 2 periods of $800. Total = 1 (CF₀) + 3 + 2 = 6 periods → B ✅

• Q5: NPV > 0 ⟺ IRR > r (for conventional projects). Sum of future CFs > P doesn't guarantee discounted sum exceeds P (A wrong). Year 1 > P is insufficient (B wrong). IRR > 0 doesn't mean it exceeds required return (C wrong) → D ✅

• Q6: CF₀ = +5,000 (received today, not discounted). Years 1-3 each +2,000 = ordinary annuity PV. Total PV = 5,000 + 2,000 × PVIFA(6%, 3) → A ✅. CF₀ is added directly in PV calculations — no discounting!