📖 时间价值 TVM（五）

📌 Topic: Solving for I/Y & N + Introduction to Irregular Cash Flows

I. Review

Today's reverse problems: 1. Given PV, FV, PMT, N → find interest rate (I/Y) 2. Given PV, FV, PMT, I/Y → find number of periods (N) 3. What if cash flows are irregular? (Paving the way for NPV)

II. Solving for Interest Rate (I/Y)

Scenario 1: Annualized Return on a Single Investment

Example 1 You invested $10,000 five years ago; current market value is $16,105. What is the annualized return?

Given: N=5, PV=−10,000, FV=16,105, PMT=0
Find: I/Y

Calculator (END mode):
N=5, PV=−10,000, PMT=0, FV=16,105 → CPT I/Y = 10%

📌 Verification: $10,000 × (1.10)⁵ = $10,000 × 1.6105 = $16,105 ✅

Scenario 2: Implied Interest Rate on an Annuity

Example 2 You borrow $20,000 to buy a car, repaying $500/month for 48 months. What is the effective annual rate?

Given: N=48, PV=20,000, PMT=−500, FV=0
Find: I/Y

Calculator:
N=48, PV=20,000, PMT=−500, FV=0 → CPT I/Y = 0.77% (monthly)

Annual rate = 0.77% × 12 ≈ 9.24%
(Effective annual rate = (1.0077)^12 − 1 ≈ 9.63%; the CFA exam typically provides answer choices)

Scenario 3: Growth Rate Interpretation

Example 3 An investment grows from $50,000 to $280,000 over 20 years with no additional contributions. Annualized growth rate?

N=20, PV=−50,000, PMT=0, FV=280,000
→ CPT I/Y = 9.03%

📌 Key insight: I/Y is not just "interest rate"—it can also be growth rate, rate of return, or discount rate depending on context.

III. Solving for Number of Periods (N)

Scenario 1: How Long to Double?

Example 4 At 7% annual interest, how long will it take $100,000 to double to $200,000?

I/Y=7, PV=−100,000, PMT=0, FV=200,000
→ CPT N = 10.24 years

📌 Rule of 72 estimate: 72 ÷ 7 = 10.3 years ← approximation; use calculator for precision on the exam

Scenario 2: How Long to Pay Off a Loan?

Example 5 Loan of $150,000 at 6% APR (0.5% monthly compounding), paying $1,000/month. How long to pay off?

I/Y=0.5, PV=150,000, PMT=−1,000, FV=0
→ CPT N = 277.7 months ≈ 23.1 years

💡 To pay off in 15 years? Solve for PMT:

N=180, I/Y=0.5, PV=150,000, FV=0 → CPT PMT = −$1,266

Need to pay an extra $266/month

Scenario 3: How Long Will Retirement Funds Last?

Example 6 At retirement you have $800,000, earning 5% annually, withdrawing $60,000/year. How many years?

I/Y=5, PV=−800,000, PMT=60,000, FV=0
→ CPT N = 27.6 years

⚠️ If switched to annuity due (beginning-of-year withdrawals):

BGN mode: I/Y=5, PV=−800,000, PMT=60,000, FV=0
→ CPT N ≈ 24.5 years (withdrawing at beginning exhausts principal faster)

IV. Solving for PMT: How Much to Save

Example 7 (Comprehensive) You plan to have $500,000 in 20 years, you currently have $50,000, with an 8% annual return. How much additional must you contribute at each year-end?

Given: N=20, I/Y=8, PV=−50,000, FV=500,000
Find: PMT

Step 1: Calculate FV of current $50,000 in 20 years
FV₅₀ₖ = 50,000 × (1.08)^20 = 50,000 × 4.6610 = $233,050

Step 2: Shortfall = 500,000 − 233,050 = $266,950

Step 3: Shortfall covered by annuity
N=20, I/Y=8, PV=0, FV=266,950 → CPT PMT = $5,834

Or use calculator in one step:
N=20, I/Y=8, PV=−50,000, FV=500,000 → CPT PMT = −$5,834

📌 Let PV grow on its own first, then use annuity to fill the gap—two-step decomposition is a reliable cross-check

V. Irregular Cash Flows: Why We Need a New Tool

Annuity formulas require three conditions: 1. Equal payment amounts each period 2. Equal intervals between payments 3. Lasts for N periods

In the real world: - Capital project: Invest $1M → recover $200K year 1 → $350K year 2 → $600K year 3 → $150K year 4 - Amounts vary each period! ✂

→ We need the NPV method (discounted cash flow summation)

VI. Manual Calculation of PV for Irregular Cash Flows

Example 8 A project has the following expected cash flows, discount rate 10%. Find total PV:

PV = 5,000/(1.10)¹ + 8,000/(1.10)² + 3,000/(1.10)³ + 10,000/(1.10)⁴
   = 5,000/1.10 + 8,000/1.21 + 3,000/1.331 + 10,000/1.4641
   = 4,545 + 6,612 + 2,254 + 6,830
   = $20,241

📌 Core concept: Discount each cash flow individually, then sum → this is the soul of NPV

VII. Calculator CF Worksheet

Example 9: Re-solve Example 8 Using CF Worksheet

Assume initial investment CF₀ = 0:
CF₀ = 0
C01 = 5,000, F01 = 1
C02 = 8,000, F02 = 1
C03 = 3,000, F03 = 1
C04 = 10,000, F04 = 1

NPV, I=10 → CPT NPV = $20,241 ✅

Example 10: Project NPV Including Initial Investment

A project requires initial investment of $25,000. Subsequent cash flows: Yr 1: $8,000; Yr 2–3: $10,000 each; Yr 4: $6,000. Discount rate 12%.

CF₀ = −25,000
C01 = 8,000,  F01 = 1
C02 = 10,000, F02 = 2    ← two consecutive years of the same amount!
C03 = 6,000,  F03 = 1

NPV, I=12 → CPT NPV = ?

Manual cross-check:
NPV = −25,000 + 8,000/(1.12) + 10,000/(1.12)² + 10,000/(1.12)³ + 6,000/(1.12)⁴
    = −25,000 + 7,143 + 7,972 + 7,118 + 3,814
    = $1,047

📌 NPV > 0 → project creates value, accept

VIII. NPV Decision Rule

💡 NPV is the absolute core of CFA Level 1—it will accompany you through the entire curriculum

IX. Key Formula Cards

📝 Practice Questions

Q1

An investment grows from $10,000 to $13,310 in 3 years. Annualized return?

A. 10%
B. 11%
C. 12%
D. 13%

Q2

At 9% annual interest, approximately how many years will it take $50,000 to grow to $100,000?

A. 7.2 years
B. 8.0 years
C. 10.2 years
D. 12.0 years

Q3

For a loan of $200,000 at 6% APR (0.5% monthly rate) with $1,200 monthly payments, what does F01 mean in the CF worksheet?

A. Discount rate
B. Initial investment amount
C. Frequency—number of consecutive periods a given cash flow repeats
D. Number of annuity periods

Q4

A project: CF₀=−$50,000, C01=$18,000 (F01=3), no other cash flows. Discount rate 10%. NPV is approximately? (Given: PVIFA factor = 2.4869)

A. −$5,236
B. −$4,500
C. $4,500
D. $5,236

Q5

Which of the following is NOT a scenario requiring irregular cash flow methods?

A. Evaluating projects with different annual payment amounts
B. Finding PV of equal, equally-spaced annuity payments
C. Valuing a business with varying annual profits
D. Investment analysis with construction-phase outlays and operational-phase recoveries

✅ Answers & Explanations