📌 课题：普通年金终值与现值
一、什么是普通年金（Ordinary Annuity）
定义： 每期期末等额收付的现金流序列
| 特征 | 说明 |
|---|---|
| 金额 | 每期相等（PMT 固定） |
| 间隔 | 等间距（每期一次） |
| 时机 | 期末支付（区别于先付年金的期初） |
| 例子 | 房贷月供、债券利息、退休定期取款 |
二、普通年金终值（FV）
FVA = PMT × [ (1+r)^n − 1 ] / r
方括号内 = 年金终值因子，记作 FVIFA(r, n)
例题 1： 每年末存 $2,000，年利率 6%，连续存 5 年，5 年末账户余额？
FVA = 2,000 × [(1.06)^5 − 1] / 0.06
= 2,000 × [1.33823 − 1] / 0.06
= 2,000 × 5.6371
= $11,274.20
验算： | 年份 | 存入 | 累计利息 | 年末余额 | |------|------|----------|----------| | 1 | 2,000 | 0 | 2,000.00 | | 2 | 2,000 | 120 | 4,120.00 | | 3 | 2,000 | 247.20 | 6,367.20 | | 4 | 2,000 | 382.03 | 8,749.23 | | 5 | 2,000 | 524.95 | 11,274.18 ✅ |
三、普通年金现值（PV）
PVA = PMT × [ 1 − 1/(1+r)^n ] / r
方括号内 = 年金现值因子，记作 PVIFA(r, n)
例题 2： 你中奖每年领 $10,000，连领 10 年（年末领），贴现率 5%，现值是多少？
PVA = 10,000 × [1 − 1/(1.05)^10] / 0.05
= 10,000 × [1 − 0.6139] / 0.05
= 10,000 × 7.7217
= $77,217
📌 如果彩票公司 offer 你一次性拿 $75,000 → 不如领 10 年！$77,217 > $75,000
四、知道三个求第四个
公式变形：
| 已知 | 求 | 公式 |
|---|---|---|
| FV, r, n | PMT | PMT = FVA / FVIFA(r, n) |
| PV, r, n | PMT | PMT = PVA / PVIFA(r, n) |
| FV, PMT, n | r | 需试算或计算器 |
| FV, PMT, r | n | n = ln(1 + FV×r/PMT) / ln(1+r) |
例题 3： 5 年后需要 $50,000，年利率 7%，每年末应存多少？
PMT = 50,000 / FVIFA(7%, 5)
= 50,000 / 5.7507
= $8,694.63
五、普通年金 vs 先付年金（预告）
| 普通年金 (Ordinary) | 先付年金 (Due) | |
|---|---|---|
| 支付时间 | 期末 | 期初 |
| n 期内支付次数 | 第 1,2,…,n 期末 | 第 0,1,…,n−1 期初 |
| FV 关系 | — | FVA_due = FVA × (1+r) |
| PV 关系 | — | PVA_due = PVA × (1+r) |
📝 练习题
Q1
每年末存 $3,000，年利率 5%，存 4 年，FVIFA(5%,4)=4.3101。终值=？
A. $12,000
B. $12,930
C. $13,250
D. $13,500
Q2
一笔年金每年末付 $5,000，共 8 年，贴现率 6%，PVIFA(6%,8)=6.2098。现值=？
A. $31,049
B. $34,000
C. $37,500
D. $40,000
Q3
10 年后需要 $100,000，年利率 8%，每年末应存多少？(FVIFA(8%,10)=14.4866)
A. $5,903
B. $6,903
C. $7,903
D. $8,903
Q4
一笔年金每年末付 $2,000，共 20 年，贴现率 4%，PVIFA(4%,20)=13.5903。现值=？
A. $27,181
B. $28,500
C. $31,000
D. $40,000
Q5
关于普通年金，以下哪个说法正确？
A. 支付发生在每期期初
B. 先付年金的 PV 比普通年金小
C. 普通年金终值因子 = [1 − 1/(1+r)^n] / r
D. 先付年金 FV = 普通年金 FV × (1+r)
✅ 答案与解析
| 题号 | 答案 | 解析 |
|---|---|---|
| Q1 | B | FVA = 3,000 × 4.3101 = $12,930.30 |
| Q2 | A | PVA = 5,000 × 6.2098 = $31,049 |
| Q3 | B | PMT = 100,000 / 14.4866 = $6,903.09 |
| Q4 | A | PVA = 2,000 × 13.5903 = $27,180.60 |
| Q5 | D | D正确。A：普通=期末，B：先付PV更大，C：那是PV公式不是FV |
📌 Topic: Ordinary Annuity – FV & PV
I. What Is an Ordinary Annuity?
Definition: A series of equal cash flows occurring at the end of each period.
| Feature | Detail |
|---|---|
| Payment Amount | Equal each period (fixed PMT) |
| Interval | Evenly spaced |
| Timing | End of period (vs. annuity due = beginning) |
| Examples | Mortgage payments, bond coupons, retirement withdrawals |
II. Future Value of an Ordinary Annuity
FVA = PMT × [(1 + r)^n − 1] / r
The bracket term is the FVIFA(r, n) — Future Value Interest Factor of an Annuity.
Example 1: Deposit $2,000 at the end of each year for 5 years at 6%. What's the FV?
FVA = 2,000 × [(1.06)^5 − 1] / 0.06
= 2,000 × [1.33823 − 1] / 0.06
= 2,000 × 5.6371
= $11,274.20
| Year | Deposit | Interest | Year-End Balance |
|---|---|---|---|
| 1 | 2,000 | 0 | 2,000.00 |
| 2 | 2,000 | 120 | 4,120.00 |
| 3 | 2,000 | 247.20 | 6,367.20 |
| 4 | 2,000 | 382.03 | 8,749.23 |
| 5 | 2,000 | 524.95 | $11,274.18 ✓ |
III. Present Value of an Ordinary Annuity
PVA = PMT × [1 − 1/(1 + r)^n] / r
The bracket term is PVIFA(r, n) — Present Value Interest Factor of an Annuity.
Example 2: You win a lottery paying $10,000/year for 10 years (end of year). Discount rate = 5%. What is the PV?
PVA = 10,000 × [1 − 1/(1.05)^10] / 0.05
= 10,000 × [1 − 0.6139] / 0.05
= 10,000 × 7.7217
= $77,217
📌 A lump-sum offer of $75,000? Reject! PVA = $77,217 > $75,000.
IV. Solving for PMT, r, or n
| Known | Solve For | Formula |
|---|---|---|
| FV, r, n | PMT | PMT = FVA / FVIFA(r, n) |
| PV, r, n | PMT | PMT = PVA / PVIFA(r, n) |
| FV, PMT, n | r | Trial & error or calculator |
| FV, PMT, r | n | n = ln(1 + FV·r/PMT) / ln(1+r) |
Example 3: You need $50,000 in 5 years. r = 7%. Annual deposit?
PMT = 50,000 / FVIFA(7%, 5) = 50,000 / 5.7507 = $8,694.63
V. Ordinary vs. Annuity Due (Preview)
| Ordinary Annuity | Annuity Due | |
|---|---|---|
| Payment timing | End of period | Beginning of period |
| FV relationship | — | FVA_due = FVA × (1+r) |
| PV relationship | — | PVA_due = PVA × (1+r) |
📌 Annuity due always has HIGHER FV and PV (one extra compounding period per payment).
📝 Practice Questions
Q1
Deposit $3,000 at year-end for 4 years at 5%. FVIFA(5%,4) = 4.3101. FV = ?
A. $12,000
B. $12,930
C. $13,250
D. $13,500
Q2
An annuity pays $5,000/year for 8 years (end of year). r = 6%. PVIFA(6%,8) = 6.2098. PV = ?
A. $31,049
B. $34,000
C. $37,500
D. $40,000
Q3
You need $100,000 in 10 years. r = 8%. FVIFA(8%,10) = 14.4866. Annual deposit = ?
A. $5,903
B. $6,903
C. $7,903
D. $8,903
Q4
An annuity pays $2,000/year for 20 years (end of year). r = 4%. PVIFA(4%,20) = 13.5903. PV = ?
A. $27,181
B. $28,500
C. $31,000
D. $40,000
Q5
Which statement about ordinary annuities is correct?
A. Payments occur at the beginning of each period
B. Annuity due has a smaller PV than ordinary annuity
C. The FV factor formula is [1 − 1/(1+r)^n] / r
D. Annuity due FV = Ordinary annuity FV × (1+r)
✅ Answer Key
| Q | Answer | Calculation |
|---|---|---|
| Q1 | B | FVA = 3,000 × 4.3101 = $12,930.30 |
| Q2 | A | PVA = 5,000 × 6.2098 = $31,049 |
| Q3 | B | PMT = 100,000 / 14.4866 = $6,903.09 |
| Q4 | A | PVA = 2,000 × 13.5903 = $27,180.60 |
| Q5 | D | D is correct. A: ordinary = end of period. B: annuity due has HIGHER PV. C: that's the PV factor, not FV. |