📌 课题：有效年利率与复利频率
一、名义利率 vs 有效年利率
| 概念 | 公式 | 说明 |
|---|---|---|
| rₛ (stated rate) | 年名义利率 | 合同或广告中报出的利率（不反映复利频率） |
| EAR | (1 + rₛ/m)^m − 1 | 真实年化收益率，考虑了年内复利 |
核心：同一个名义利率，复利频率越高，EAR 越大
二、EAR 计算示例
例题 1： 某银行提供年利率 8%，分别按年、半年、季、月、连续复利，求 EAR。
| 复利频率 | m | EAR |
|---|---|---|
| 年 | 1 | (1 + 0.08/1)^1 − 1 = 8.00% |
| 半年 | 2 | (1 + 0.08/2)^2 − 1 = 8.16% |
| 季 | 4 | (1 + 0.08/4)^4 − 1 = 8.24% |
| 月 | 12 | (1 + 0.08/12)^12 − 1 = 8.30% |
| 连续 | ∞ | EAR = e^0.08 − 1 = 8.33% |
📌 月复利的 EAR (8.30%) 比名义利率高 0.30%——这就是"复利效应"
三、连续复利下的 EAR
| 关系 | 公式 |
|---|---|
| EAR（连续） | EAR = e^(rₛ) − 1 |
| 连续 FV | FV = PV × e^(rₛ × n) |
| 连续 PV | PV = FV / e^(rₛ × n) |
四、EAR 转换与比较
场景： 你要存款，三家银行报价： - A 银行：名义 7.8%，季复利 - B 银行：名义 7.7%，月复利 - C 银行：名义 7.9%，半年复利
| 银行 | 计算 | EAR |
|---|---|---|
| A | (1+0.078/4)^4 − 1 | 8.03% |
| B | (1+0.077/12)^12 − 1 | 7.98% |
| C | (1+0.079/2)^2 − 1 | 8.06% ✅ |
📌 结论：C 银行最优！ 名义利率最低 ≠ EAR 最低，必须统一换算后比较
五、期间利率（Periodic Rate）
| 概念 | 计算 |
|---|---|
| 期间利率 | rₛ / m |
| 月利率 | 年名义利率 / 12 |
| 日利率 | 年名义利率 / 365 |
| 总期数 | 年数 × m |
例题 2： 年名义利率 12%，月复利，存 2 年
月利率 = 12% / 12 = 1% = 0.01
总期数 = 2 × 12 = 24
FV = PV × (1.01)^24
六、FV 与 PV 的复利频率效应总结
| 增大复利频率 (m↑) | 对 FV 的影响 | 对 PV 的影响 |
|---|---|---|
| 给定 PV、r、n | FV ↑ | — |
| 给定 FV、r、n | — | PV ↓ |
直观理解： 复利越频繁，钱"滚"得越快，所以终值更大 / 现值更小
📝 练习题
Q1
名义年利率 10%，半年复利，EAR 最接近？
A. 10.00%
B. 10.25%
C. 10.38%
D. 10.50%
Q2
银行 A：名义 6.0% 年复利 vs 银行 B：名义 5.9% 月复利。哪个 EAR 更高？
A. 银行 A
B. 银行 B
C. 一样
D. 无法判断
Q3
连续复利下，名义利率 5%，EAR = ?
A. 5.00%
B. 5.13%
C. 5.20%
D. 5.52%
Q4
$1,000 以名义年利率 12%、季复利存 1 年，终值是多少？
A. $1,120
B. $1,125
C. $1,126
D. $1,127
Q5
一笔债务 5 年后需偿还 $10,000，贴现率 9%、连续复利，现在需要准备多少？
A. $6,376
B. $6,420
C. $6,499
D. $6,531
✅ 答案与解析
| 题号 | 答案 | 解析 |
|---|---|---|
| Q1 | B | EAR = (1+0.10/2)^2 − 1 = 1.1025 − 1 = 10.25% |
| Q2 | B | A: EAR=6.00% |
| Q3 | B | EAR = e^0.05 − 1 = 1.05127 − 1 = 5.13% |
| Q4 | C | FV = 1,000 × (1.03)^4 = 1,000 × 1.125509 = $1,125.51 ≈ $1,126 |
| Q5 | A | PV = 10,000 / e^(0.09×5) = 10,000 / 1.5683 = $6,376.28 |
Module 2: Quantitative Methods — Time Value of Money (TVM) Basics Progress: 88/560 lessons Date: 2026-06-16
🔄 Review (L087 Recap)
- Time Value of Money (TVM): A dollar today is worth more than a dollar tomorrow
- Future Value (FV): FV = PV × (1 + r)^n — the value of an investment after n periods at rate r
- Present Value (PV): PV = FV / (1 + r)^n — today's value of a future cash flow
- Interest Rate (r): Also called discount rate, required rate of return, or opportunity cost
📘 New Concepts
1. Compound Interest vs. Simple Interest
Simple Interest: Interest is earned only on the original principal.
FV = PV × (1 + r × n)
Compound Interest: Interest is earned on both the original principal AND previously accumulated interest.
FV = PV × (1 + r)^n
Key insight: With compound interest, money grows exponentially. The longer the horizon, the larger the gap between compound and simple interest.
2. Compounding Frequency
When interest compounds more than once per year, we adjust the formula:
FV = PV × (1 + r_s / m)^(m × n)
Where: - r_s = stated annual interest rate (nominal rate) - m = number of compounding periods per year - n = number of years
| Compounding Frequency | m |
|---|---|
| Annual | 1 |
| Semi-annual | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
Example: $1,000 at 8% stated rate, compounded quarterly for 2 years:
FV = 1,000 × (1 + 0.08/4)^(4×2) = 1,000 × (1.02)^8 = $1,171.66
Compare with annual compounding: 1,000 × (1.08)^2 = $1,166.40 → More frequent compounding → higher FV!
3. Continuous Compounding
As m → ∞, compounding becomes continuous:
FV = PV × e^(r_s × n)
Where e ≈ 2.71828 (Euler's number).
Example: $1,000 at 8%, continuously compounded for 2 years:
FV = 1,000 × e^(0.08×2) = 1,000 × e^0.16 = 1,000 × 1.17351 = $1,173.51
4. Effective Annual Rate (EAR)
EAR tells you the ACTUAL annual return when compounding is more frequent than annual:
EAR = (1 + r_s / m)^m − 1
For continuous compounding:
EAR = e^(r_s) − 1
| Stated Rate | Compounding | EAR |
|---|---|---|
| 8% | Annual | 8.00% |
| 8% | Semi-annual | 8.16% |
| 8% | Quarterly | 8.24% |
| 8% | Monthly | 8.30% |
| 8% | Continuous | 8.33% |
Key: As m increases, EAR increases — but at a decreasing rate (diminishing marginal effect).
5. Stated Rate vs. EAR: Solving Problems
When the question gives a "stated annual rate" with non-annual compounding, ALWAYS convert:
Method A — Use periodic rate directly:
Periodic rate = r_s / m Number of periods = m × n FV = PV × (1 + periodic rate)^(total periods)
Method B — Use EAR with annual periods:
EAR = (1 + r_s/m)^m − 1 FV = PV × (1 + EAR)^n
Both methods give the same answer. Pick whichever is easier for you.
📐 Worked Examples
Example 1: Comparing Compounding Frequencies
You invest $5,000 for 3 years at a stated annual rate of 6%. Calculate the FV under: - (a) Annual compounding - (b) Semi-annual compounding - (c) Monthly compounding - (d) Continuous compounding
Solution:
(a) FV = 5,000 × (1.06)^3 = $5,955.08
(b) FV = 5,000 × (1 + 0.06/2)^6 = 5,000 × (1.03)^6 = $5,970.26
(c) FV = 5,000 × (1 + 0.06/12)^36 = 5,000 × (1.005)^36 = $5,983.40
(d) FV = 5,000 × e^(0.06×3) = 5,000 × e^0.18 = $5,986.09
Example 2: Finding the EAR
A bank advertises a savings account with a stated annual rate of 5%, compounded daily (365 days). What is the EAR?
Solution:
EAR = (1 + 0.05/365)^365 − 1 EAR = (1.000137)^365 − 1 EAR = 1.05127 − 1 EAR = 5.127%
Example 3: Solving for the Stated Rate
You need an EAR of at least 10%. If the bank compounds quarterly, what is the minimum stated annual rate you should accept?
Solution:
EAR = (1 + r_s/4)^4 − 1 = 0.10 (1 + r_s/4)^4 = 1.10 1 + r_s/4 = (1.10)^(1/4) = 1.02411 r_s/4 = 0.02411 r_s = 9.64%
📝 Practice Quiz
Q1. Which of the following statements about compound interest is CORRECT?
A) With compound interest, interest is earned only on the original principal B) Compound interest always yields a lower future value than simple interest C) The more frequent the compounding, the higher the effective annual rate D) Continuous compounding and annual compounding yield the same FV if the stated rate is the same
Q2. An investor deposits $10,000 in an account earning a stated annual rate of 7%, compounded monthly. After 2 years, the account balance is CLOSEST to:
A) $11,400 B) $11,498 C) $11,500 D) $11,510
Q3. A bank quotes a stated annual interest rate of 12% with quarterly compounding. The effective annual rate (EAR) is CLOSEST to:
A) 12.00% B) 12.36% C) 12.55% D) 12.68%
Q4. The difference between the effective annual rate and the stated annual rate will be LARGEST when:
A) The stated rate is low and compounding is infrequent B) The stated rate is high and compounding is frequent C) The stated rate is high and compounding is infrequent D) The stated rate is low and compounding is frequent
Q5. For a stated annual rate of 10% compounded continuously, the effective annual rate is CLOSEST to:
A) 10.00% B) 10.25% C) 10.52% D) 10.71%
📌 Summary
| Concept | Formula |
|---|---|
| Simple Interest FV | PV × (1 + r × n) |
| Compound Interest FV | PV × (1 + r)^n |
| FV with m compounding/yr | PV × (1 + r_s/m)^(m×n) |
| Continuous Compounding FV | PV × e^(r_s×n) |
| Effective Annual Rate | (1 + r_s/m)^m − 1 |
| Continuous EAR | e^(r_s) − 1 |
Golden Rules: 1. More frequent compounding → higher EAR → higher FV 2. Always convert the stated annual rate to the appropriate periodic rate before solving TVM problems 3. EAR allows you to compare investments with different compounding frequencies on an apples-to-apples basis 4. For any non-annual compounding frequency, EAR > stated rate (except when m = 1)
Next Lesson: L089 — Discounted Cash Flow (DCF) Applications