📌 课题：先付年金与永续年金
一、回顾：普通年金 vs 先付年金
| 类型 | 英文 | 现金流发生时间 | 按键（计算器） |
|---|---|---|---|
| 普通年金 | Ordinary Annuity | 期末 | END 模式 |
| 先付年金 | Annuity Due | 期初 | BGN 模式 |
关键区别： 同样是每年 $100、连付 5 年，先付年金的每次付款都"提前了一期" → 终值更大（多赚一期利息），现值也更大（少折现一期）
二、先付年金的 FV 与 PV 公式
| 公式 | 表达式 |
|---|---|
| FV(先付) | FV(普通) × (1 + r) |
| PV(先付) | PV(普通) × (1 + r) |
📌 实战心法： 先付年金 = 普通年金 × (1 + r)，不需要死记新公式！
三、先付年金例题
例题 1：终值 你每年年初存入 $2,000，年利率 6%，存 10 年，到期有多少钱？
方法一：计算器调至 BGN 模式
N=10, I/Y=6, PV=0, PMT=−2,000 → CPT FV = $27,943
方法二：先用 END 模式算普通年金，再 ×1.06
普通 FV = 2,000 × [(1.06^10 − 1)/0.06] = 2,000 × 13.1808 = $26,362
先付 FV = 26,362 × 1.06 = $27,943 ✅
例题 2：现值 一份保险合同约定每年年初支付你 $5,000，连续 15 年，贴现率 8%。该年金现值是多少？
BGN 模式：N=15, I/Y=8, PMT=5,000, FV=0 → CPT PV = $46,234
END 模式下先算普通现值：
PV(普通) = 5,000 × [1 − (1.08)^−15] / 0.08 = 5,000 × 8.5595 = $42,797
PV(先付) = 42,797 × 1.08 = $46,221 ≈ $46,234
📌 先付年金现值更高——钱更早拿到手，当然更值钱
四、永续年金（Perpetuity）
| 概念 | 说明 |
|---|---|
| 定义 | 无限期支付固定金额的年金（n = ∞） |
| 典型例子 | 优先股股息、英国永续债券（Consols） |
| 前提 | 每期金额相同、间隔相等、永不休止 |
🔑 永续年金现值公式
PV(永续) = PMT / r
推导直觉： 如果你想每年永远领 $100，而利率是 5%，你需要存 $100/0.05 = $2,000 本金来"生出"这笔利息
例题 3：永续年金现值
某优先股每年派息 $4.50，要求回报率 9%，该股票值多少钱？
PV = 4.50 / 0.09 = $50.00
例题 4：增长率永续年金（Growing Perpetuity）
某股票刚派发股息 $2.00，预计每年增长 3%，要求回报率 10%，估值多少？
PV = PMT₁ / (r − g) = D₀×(1+g) / (r − g)
PV = 2.00 × 1.03 / (0.10 − 0.03)
PV = 2.06 / 0.07 = $29.43
⚠️ 前提：r > g，否则模型失效
五、三者对比总表
| 类型 | 现金流特征 | PV 公式 | FV 公式 |
|---|---|---|---|
| 普通年金 | 期末支付、有限期 | PMT × [(1−(1+r)^−n)/r] | PMT × [((1+r)^n−1)/r] |
| 先付年金 | 期初支付、有限期 | 普通 PV × (1+r) | 普通 FV × (1+r) |
| 永续年金 | 期末支付、无限期 | PMT / r | 无（n→∞） |
六、实战应用：退休规划
你 30 岁，计划从 60 岁起每年年初领 $50,000 退休金，领 25 年。假设年利率 7%：
Step 1： 60 岁时需要多少钱？（先付年金的 PV）
N=25, I/Y=7, PMT=50,000, FV=0, BGN → CPT PV = $623,589
Step 2： 从 30 岁到 60 岁，每年年末要存多少？
N=30, I/Y=7, PV=0, FV=623,589 → CPT PMT = $6,314
📌 每年存 $6,314，30 年后可实现退休目标——这就是 TVM 的力量
📝 练习题
Q1
普通年金 FV = $10,000，若改为先付年金（其他条件不变），FV 为？
A. $10,000
B. $10,000 × r
C. $10,000 × (1 + r)
D. $10,000 / (1 + r)
Q2
某永续年金每年支付 $500，贴现率 10%，现值为？
A. $500
B. $5,000
C. $50,000
D. 无穷大
Q3
一笔先付年金：年付 $1,000、利率 5%、共 3 年。第一笔今天付。终值等于？
（已知普通年金 FV因子 = 3.1525）
A. $3,153
B. $3,310
C. $3,152
D. $3,471
Q4
某增长型永续年金：下一期支付 $100，增长率 4%，贴现率 9%，现值为？
A. $1,111
B. $2,000
C. $2,500
D. $1,000
Q5
以下哪种情况不适合用永续年金公式估值？
A. 优先股（无到期日、固定股息）
B. 英国 Consols 债券
C. 一只预期永续经营的成熟公司股票
D. 一张 10 年到期的固定利率债券
✅ 答案与解析
| 题号 | 答案 | 解析 |
|---|---|---|
| Q1 | C | 先付年金每笔多赚一期利息，FV(先付) = FV(普通) × (1+r) |
| Q2 | B | PV = 500 / 0.10 = $5,000 |
| Q3 | B | FV(普通) = 1,000 × 3.1525 = 3,152.5 → FV(先付) = 3,152.5 × 1.05 = $3,310.13 ≈ $3,310 |
| Q4 | B | PV = 100 / (0.09 − 0.04) = 100 / 0.05 = $2,000 |
| Q5 | D | 10 年到期债券是有限期现金流，应用普通年金 PV 公式，永续年金假设 n→∞ |
📌 Topic: Annuity Due & Perpetuity
I. Review: Ordinary Annuity vs. Annuity Due
| Type | Cash Flow Timing | Calculator Mode |
|---|---|---|
| Ordinary Annuity | End of period | END mode |
| Annuity Due | Beginning of period | BGN mode |
Key difference: Same $100/year for 5 years — annuity due payments arrive one period earlier → Higher FV (earns one extra compounding period), higher PV (one less discounting period)
II. Annuity Due: FV & PV Formulas
| Formula | Expression |
|---|---|
| FV (Annuity Due) | FV (Ordinary) × (1 + r) |
| PV (Annuity Due) | PV (Ordinary) × (1 + r) |
📌 Pro tip: Annuity Due = Ordinary Annuity × (1 + r) — no need to memorize new formulas!
III. Annuity Due Examples
Example 1: Future Value You deposit $2,000 at the beginning of each year for 10 years at 6%. What is the FV?
Method 1: Switch calculator to BGN mode
N=10, I/Y=6, PV=0, PMT=−2,000 → CPT FV = $27,943
Method 2: Compute ordinary annuity in END mode, then ×1.06
Ordinary FV = 2,000 × [(1.06^10 − 1)/0.06] = 2,000 × 13.1808 = $26,362
Annuity Due FV = 26,362 × 1.06 = $27,943 ✓
Example 2: Present Value An insurance contract pays you $5,000 at the beginning of each year for 15 years. Discount rate = 8%. What is the PV?
BGN mode: N=15, I/Y=8, PMT=5,000, FV=0 → CPT PV = $46,234
END mode: compute ordinary PV first:
PV (Ordinary) = 5,000 × [1 − (1.08)^−15] / 0.08 = 5,000 × 8.5595 = $42,797
PV (Annuity Due) = 42,797 × 1.08 = $46,221 ≈ $46,234
📌 Annuity due PV is higher — money received earlier is worth more
IV. Perpetuity
| Concept | Explanation |
|---|---|
| Definition | An annuity with infinite payments (n = ∞) |
| Classic examples | Preferred stock dividends, UK Consols |
| Conditions | Equal payments, equal intervals, no end date |
🔑 Perpetuity PV Formula
PV (Perpetuity) = PMT / r
Intuition: To receive $100/year forever at 5%, you need $100/0.05 = $2,000 in principal to "generate" the interest stream.
Example 3: Perpetuity Valuation
A preferred stock pays $4.50 annual dividend. Required return = 9%. What is it worth?
PV = 4.50 / 0.09 = $50.00
Example 4: Growing Perpetuity
A stock just paid a $2.00 dividend, expected to grow 3%/year. Required return = 10%. Valuation?
PV = PMT₁ / (r − g) = D₀ × (1+g) / (r − g)
PV = 2.00 × 1.03 / (0.10 − 0.03)
PV = 2.06 / 0.07 = $29.43
⚠️ Requirement: r > g, or the model breaks down
V. Comparison: All Three Types
| Type | Cash Flow Pattern | PV Formula | FV Formula |
|---|---|---|---|
| Ordinary Annuity | End-of-period, finite n | PMT × [(1−(1+r)^−n)/r] | PMT × [((1+r)^n−1)/r] |
| Annuity Due | Beginning-of-period, finite n | Ordinary PV × (1+r) | Ordinary FV × (1+r) |
| Perpetuity | End-of-period, infinite n | PMT / r | None (n→∞) |
VI. Application: Retirement Planning
You're 30. From age 60, you want $50,000/year at the beginning of each year for 25 years. Assume r = 7%.
Step 1: How much capital needed at age 60? (PV of annuity due)
N=25, I/Y=7, PMT=50,000, FV=0, BGN → CPT PV = $623,589
Step 2: How much to save at each year-end from age 30 to 60?
N=30, I/Y=7, PV=0, FV=623,589 → CPT PMT = $6,314
📌 Save $6,314/year for 30 years → retirement goal achieved. That's the power of TVM.
📝 Practice Questions
Q1
The FV of an ordinary annuity = $10,000. If switched to an annuity due (all else equal), the FV is:
A. $10,000
B. $10,000 × r
C. $10,000 × (1 + r)
D. $10,000 / (1 + r)
Q2
A perpetuity pays $500/year. Discount rate = 10%. PV = ?
A. $500
B. $5,000
C. $50,000
D. Infinite
Q3
Annuity due: $1,000/year, r = 5%, n = 3 years. First payment today. FV = ? (Ordinary annuity FV factor = 3.1525)
A. $3,153
B. $3,310
C. $3,152
D. $3,471
Q4
A growing perpetuity: next payment = $100, g = 4%, r = 9%. PV = ?
A. $1,111
B. $2,000
C. $2,500
D. $1,000
Q5
Which situation is NOT appropriate for perpetuity valuation?
A. Preferred stock (no maturity, fixed dividend)
B. UK Consols
C. A mature company stock expected to operate indefinitely
D. A 10-year fixed-rate bond
✅ Answer Key
| Q | Answer | Explanation |
|---|---|---|
| Q1 | C | Annuity due earns one extra compounding period: FV(due) = FV(ordinary) × (1+r) |
| Q2 | B | PV = 500 / 0.10 = $5,000 |
| Q3 | B | FV(ordinary) = 1,000 × 3.1525 = 3,152.5 → FV(due) = 3,152.5 × 1.05 = $3,310.13 ≈ $3,310 |
| Q4 | B | PV = 100 / (0.09 − 0.04) = 100 / 0.05 = $2,000 |
| Q5 | D | A 10-year bond has finite cash flows → use ordinary annuity PV formula. Perpetuity assumes n→∞. |