📌 课题：当复利不再按月、按天，而是每分每秒都在滚动
一、从离散到连续：一个思维跳跃
L095 我们学了：复利频率越高 → EAR 越大。
| 名义 12%，复利频率 | EAR |
|---|---|
| 年（m=1） | 12.000% |
| 半年（m=2） | 12.360% |
| 季（m=4） | 12.551% |
| 月（m=12） | 12.683% |
| 日（m=365） | 12.747% |
| 时（m=8760） | 12.750% |
问题来了：如果 m → ∞（每时每刻都在复利），EAR 会无限大吗？
答案是否定的。它收敛于一个有限值。
二、数学推导：从极限到 e
2.1 极限表达式
离散复利的 EAR 公式：
$$EAR = \left(1 + \frac{r_s}{m}\right)^m - 1$$
让 $m \to \infty$：
$$\lim_{m \to \infty} \left(1 + \frac{r_s}{m}\right)^m = e^{r_s}$$
🔑 关键极限： $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.718281828$
将 $n = m/r_s$ 代入即得。
2.2 连续复利的 EAR
$$\boxed{EAR_{continuous} = e^{r_s} - 1}$$
2.3 连续复利的 FV / PV
$$\boxed{FV_n = PV \times e^{r_s \times n}}$$
$$\boxed{PV = FV_n \times e^{-r_s \times n}}$$
🔑 $e^{r_s \times n}$ 就是连续复利的终值因子。它替代了离散复利中的 $(1 + r)^n$。
三、e 的直觉理解
3.1 e 不是凭空捏造的数字
$e$ 来源于极限复利这个自然的金融过程：
存 1 元，年利率 100%：
年复利 1 次 → (1 + 1.00)¹ = 2.00000
半年复利 2 次 → (1 + 0.50)² = 2.25000
季复利 4 次 → (1 + 0.25)⁴ = 2.44141
月复利 12 次 → (1 + 1/12)¹² = 2.61304
日复利 365 次 → (1+1/365)³⁶⁵ = 2.71457
。。。
→ m→∞ → e ≈ 2.71828
💡 100% 年利率、连续复利 → 1 元变成 $2.71828，不会超过 e。
3.2 连续复利的增长率
连续复利的增长是指数级的，且增长速度为 $r_s$：
$$\frac{dV}{dt} = r_s \times V(t)$$
这个微分方程的解就是 $V(t) = V_0 \times e^{r_s t}$。
四、连续复利在 CFA 中的三大应用场景
场景 1：理论定价模型（衍生品）
Black-Scholes 期权定价模型的核心假设：股价服从连续复利的几何布朗运动。
$$S_T = S_0 \times e^{(r - \sigma^2/2)T + \sigma \sqrt{T} \cdot Z}$$
连续复利 $r$ 是 B-S 模型的基本输入。
🔑 CFA 一级衍生品 / 二级定量方法会反复用到连续复利。
场景 2：利率期限结构（固定收益）
零息债券的即期利率（Spot Rate）通常以连续复利报价：
$$P = F \times e^{-r \times T}$$
其中 $P$ = 债券现价，$F$ = 面值，$r$ = 连续复利即期利率，$T$ = 年限。
场景 3：对数收益率（Log Return）
连续复利收益率 = 对数收益率：
$$\boxed{r_{continuous} = \ln\left(\frac{P_t}{P_{t-1}}\right) = \ln(1 + R)}$$
其中 $R$ = 简单收益率（holding period return）。
对数收益率的优势： - ✅ 可加性： 多期连续收益率可以直接相加 - ✅ 对称性： 涨 10% 再跌 10%，对数收益率之和 = 0 - ✅ 正态性假设： 对数收益率比简单收益率更接近正态分布
| 日期 | 价格 | 简单收益率 | 对数收益率 |
|---|---|---|---|
| Day 0 | $100 | — | — |
| Day 1 | $110 | +10.00% | ln(1.10) = +9.53% |
| Day 2 | $99 | −10.00% | ln(0.90) = −10.54% |
| 累计 | — | −1.00% | −1.01% |
💡 对数收益率相加 = 9.53% + (−10.54%) = −1.01%，直接对应 $100 → $99 的连续收益率。
五、连续复利与离散复利的换算
5.1 等价关系
给定名义年利率 $r_s$，连续复利 $r_c$：
$$\boxed{r_c = \ln\left(1 + EAR\right) = \ln\left[\left(1 + \frac{r_s}{m}\right)^m\right] = m \times \ln\left(1 + \frac{r_s}{m}\right)}$$
反过来：
$$\boxed{r_s = m \times \left(e^{r_c / m} - 1\right)}$$
5.2 实例
例 1： 名义 8%，半年复利 → 等价的连续复利利率？
$$EAR = (1.04)^2 - 1 = 8.16\%$$ $$r_c = \ln(1.0816) = 7.844\%$$
验算：$e^{0.07844} - 1 = 1.0816 - 1 = 8.16\%$ ✅
例 2： 连续复利 6% → 等价的季复利名义利率？
$$EAR = e^{0.06} - 1 = 6.1837\%$$ $$r_s = 4 \times [(1.061837)^{1/4} - 1] = 4 \times [1.01511 - 1] = 6.044\%$$
🔑 连续复利 6% ≈ 季复利名义 6.044%，两者 EAR 相同。
六、连续复利的 TVM 完整运算
6.1 单笔现金流
| 方向 | 离散公式 | 连续公式 |
|---|---|---|
| FV | $PV(1+r)^n$ | $PV \times e^{r_s n}$ |
| PV | $FV/(1+r)^n$ | $FV \times e^{-r_s n}$ |
6.2 年金：连续复利 + 等额支付
当年金支付以连续复利折现时：
$$\boxed{PV_{annuity} = PMT \times \frac{1 - e^{-r \times n}}{e^r - 1}}$$
$$\boxed{FV_{annuity} = PMT \times \frac{e^{r \times n} - 1}{e^r - 1}}$$
⚠️ 这个公式 CFA 一级很少要求手算，但理解推导逻辑很重要。
6.3 计算实例
案例： 每年末存 $1,000，连续复利 5%，存 3 年，终值？
$$FV = 1{,}000 \times \frac{e^{0.05 \times 3} - 1}{e^{0.05} - 1}$$
先算因子： - $e^{0.15} = 1.16183$ - $e^{0.05} = 1.05127$
$$FV = 1{,}000 \times \frac{1.16183 - 1}{1.05127 - 1} = 1{,}000 \times \frac{0.16183}{0.05127} = \$3{,}156.40$$
对比离散年复利 5%：$1{,}000 \times \frac{(1.05)^3 - 1}{0.05} = \$3{,}152.50$
七、金融计算器操作（BA II Plus）
7.1 计算 $e^x$
BA II Plus 没有直接的 $e^x$ 键，但可以用 LN 的反函数：
输入 x → [2nd] [LN]（即 e^x）
例如计算 $e^{0.15}$：
0.15 [2nd] [LN] → 1.161834
7.2 连续复利 FV
例： PV = 10,000，r = 6%，n = 5 年，连续复利。
步骤 1：计算 r × n = 0.06 × 5 = 0.30
步骤 2：0.30 [2nd] [LN] → e^0.30 = 1.349859
步骤 3：× 10,000 = 13,498.59
7.3 连续复利 PV
例： FV = 20,000，r = 8%，n = 3 年。
步骤 1：计算 r × n = 0.08 × 3 = 0.24
步骤 2：0.24 [+/-] [2nd] [LN] → e^(-0.24) = 0.786628
步骤 3：× 20,000 = 15,732.56
八、常见陷阱
| 陷阱 | 正解 |
|---|---|
| 把连续复利的 $r$ 当离散利率用 | $e^{r}$ ≠ $1+r$，用正确公式 |
| 忽略 $e^{r_s \times n}$ 中的 n | 指数是 $r_s \times n$（年数），不是 $r_s$ |
| 对数收益率直译成简单收益率 | $r = e^{r_c} - 1$ 才能回推 |
| 连续年金公式套用离散因子 | 分母是 $e^r - 1$，不是 $r$ |
| $e$ 的近似值用 2.7 | CFA 计算精确到 4 位小数，用计算器的 $e^x$ |
九、CFA 典型考题
题 1（连续复利 FV）
$8,000 以连续复利年利率 7% 投资 4 年，终值最接近：
A. $10,488 B. $10,563 C. $10,600
题 2（连续复利 PV）
5 年后需要 $50,000，连续复利年利率 5%，现在需存入多少？
A. $38,940 B. $39,175 C. $39,400
题 3（EAR 比较）
以下哪个 EAR 最大？
A. 名义 10%，年复利 B. 名义 9.8%，月复利 C. 名义 9.5%，连续复利
题 4（连续复利 ← → 离散换算）
某投资的连续复利年收益率为 8.00%，等效的季复利名义年利率最接近：
A. 8.00% B. 8.08% C. 8.16%
题 5（对数收益率）
某股票价格从 $50 涨到 $55，连续复利收益率（对数收益率）最接近：
A. 9.53% B. 10.00% C. 10.54%
题 6（极限概念）
名义年利率 10%，当 m → ∞（连续复利），1 元 1 年后的终值最接近：
A. 1.1000 B. 1.1052 C. 1.1100
十、答案
| 题号 | 答案 | 解析 |
|---|---|---|
| 1 | B | $FV = 8{,}000 \times e^{0.07 \times 4} = 8{,}000 \times e^{0.28} = 8{,}000 \times 1.32313 = 10{,}585$ ≈ $10,563 |
| 2 | A | $PV = 50{,}000 \times e^{-0.05 \times 5} = 50{,}000 \times e^{-0.25} = 50{,}000 \times 0.77880 = 38{,}940$ |
| 3 | C | A: 10.00%；B: $(1+0.098/12)^{12} - 1 = 10.25\%$；C: $e^{0.095} - 1 = 9.966\%$。B 最大！ |
| 4 | B | $EAR = e^{0.08} - 1 = 8.329\%$；$r_s = 4 \times [(1.08329)^{1/4} - 1] = 4 \times 0.0202 = 8.08\%$ |
| 5 | A | $r_c = \ln(55/50) = \ln(1.10) = 0.09531 = 9.53\%$ |
| 6 | B | $(1 + 0.10/m)^m$ 当 $m \to \infty$ → $e^{0.10} = 1.105171$ ≈ 1.1052 |
十一、公式速查表
| 公式 | 表达式 |
|---|---|
| 连续复利 EAR | $EAR = e^{r_s} - 1$ |
| 连续复利 FV | $FV = PV \times e^{r_s \times n}$ |
| 连续复利 PV | $PV = FV \times e^{-r_s \times n}$ |
| 连续 → 离散名义（m 次） | $r_s = m \times (e^{r_c/m} - 1)$ |
| 离散 → 连续 | $r_c = m \times \ln(1 + r_s/m)$ |
| 对数收益率 | $r_{log} = \ln(P_t / P_{t-1}) = \ln(1+R)$ |
| 连续年金 PV | $PV = PMT \times \frac{1 - e^{-r \times n}}{e^r - 1}$ |
| 连续年金 FV | $FV = PMT \times \frac{e^{r \times n} - 1}{e^r - 1}$ |
| 连续复利折现因子 | $e^{-r \times n}$ |
十二、CFA 考试关联
| 项目 | 详情 |
|---|---|
| CFA 科目 | Quantitative Methods |
| 关联章节 | TVM → 连续复利 / 利率类型 |
| 前置知识 | L089–L094（PV/FV/年金）、L095（名义 vs EAR） |
| 后续关联 | 固定收益（即期利率连续复利报价）、衍生品（B-S 模型）、组合管理（对数收益） |
| 考试形式 | 计算题为主（FV/PV/EAR/换算） |
📊 核心信条：连续复利是离散复利的理论极限。$e^{r_s}$ 替代 $(1+r)^n$，对数收益率替代简单收益率。衍生品和固定收益定价中，连续复利是默认语言。
📌 Topic: When Compounding Is No Longer Monthly or Daily, But Every Instant
1. From Discrete to Continuous: A Mental Leap
In L095, we learned: higher compounding frequency → higher EAR.
| Nominal 12%, Compounding Frequency | EAR |
|---|---|
| Annual (m=1) | 12.000% |
| Semiannual (m=2) | 12.360% |
| Quarterly (m=4) | 12.551% |
| Monthly (m=12) | 12.683% |
| Daily (m=365) | 12.747% |
| Hourly (m=8,760) | 12.750% |
The question: If m → ∞ (compounding every instant), does EAR go to infinity?
The answer is no. It converges to a finite limit.
2. Mathematical Derivation: From Limits to e
2.1 The Limit Expression
Discrete compounding EAR formula:
$$EAR = \left(1 + \frac{r_s}{m}\right)^m - 1$$
Let $m \to \infty$:
$$\lim_{m \to \infty} \left(1 + \frac{r_s}{m}\right)^m = e^{r_s}$$
🔑 Key limit: $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.718281828$
Substitute $n = m/r_s$ to derive.
2.2 EAR Under Continuous Compounding
$$\boxed{EAR_{continuous} = e^{r_s} - 1}$$
2.3 FV / PV Under Continuous Compounding
$$\boxed{FV_n = PV \times e^{r_s \times n}}$$
$$\boxed{PV = FV_n \times e^{-r_s \times n}}$$
🔑 $e^{r_s \times n}$ is the future value factor for continuous compounding. It replaces $(1 + r)^n$ from discrete compounding.
3. Building Intuition for e
3.1 e Is Not an Arbitrary Number
$e$ arises naturally from the process of limit compounding in finance:
Invest $1 at 100% annual rate:
Annual compounding (1x) → (1 + 1.00)¹ = 2.00000
Semiannual (2x) → (1 + 0.50)² = 2.25000
Quarterly (4x) → (1 + 0.25)⁴ = 2.44141
Monthly (12x) → (1 + 1/12)¹² = 2.61304
Daily (365x) → (1 + 1/365)³⁶⁵ = 2.71457
...
→ m → ∞ → e ≈ 2.71828
💡 At 100% continuously compounded, $1 grows to $2.71828 — never exceeding e.
3.2 The Growth Rate Under Continuous Compounding
Continuous compounding produces exponential growth, with instantaneous growth rate $r_s$:
$$\frac{dV}{dt} = r_s \times V(t)$$
The solution to this differential equation is $V(t) = V_0 \times e^{r_s t}$.
4. Three Major Applications in the CFA Curriculum
Application 1: Theoretical Pricing Models (Derivatives)
The Black-Scholes option pricing model assumes: stock prices follow geometric Brownian motion with continuous compounding.
$$S_T = S_0 \times e^{(r - \sigma^2/2)T + \sigma \sqrt{T} \cdot Z}$$
The continuously compounded rate $r$ is a fundamental input to the B-S model.
🔑 Continuous compounding appears repeatedly in CFA Level I Derivatives and Level II Quantitative Methods.
Application 2: Interest Rate Term Structure (Fixed Income)
Zero-coupon bond spot rates are often quoted on a continuously compounded basis:
$$P = F \times e^{-r \times T}$$
Where $P$ = bond price, $F$ = face value, $r$ = continuously compounded spot rate, $T$ = time to maturity.
Application 3: Log Returns
Continuously compounded return = log return:
$$\boxed{r_{continuous} = \ln\left(\frac{P_t}{P_{t-1}}\right) = \ln(1 + R)}$$
Where $R$ = simple holding period return.
Advantages of log returns: - ✅ Additivity: Multi-period continuous returns can be directly summed - ✅ Symmetry: +10% then −10% → log returns sum to zero - ✅ Normality assumption: Log returns are closer to normal distribution than simple returns
| Date | Price | Simple Return | Log Return |
|---|---|---|---|
| Day 0 | $100 | — | — |
| Day 1 | $110 | +10.00% | ln(1.10) = +9.53% |
| Day 2 | $99 | −10.00% | ln(0.90) = −10.54% |
| Cumulative | — | −1.00% | −1.01% |
💡 Sum of log returns = 9.53% + (−10.54%) = −1.01%, directly corresponding to the continuous return from $100 → $99.
5. Converting Between Continuous and Discrete Compounding
5.1 Equivalence Relationships
Given nominal annual rate $r_s$, the equivalent continuous rate $r_c$:
$$\boxed{r_c = \ln\left(1 + EAR\right) = \ln\left[\left(1 + \frac{r_s}{m}\right)^m\right] = m \times \ln\left(1 + \frac{r_s}{m}\right)}$$
Conversely:
$$\boxed{r_s = m \times \left(e^{r_c / m} - 1\right)}$$
5.2 Examples
Example 1: Nominal 8%, semiannual compounding → equivalent continuous rate?
$$EAR = (1.04)^2 - 1 = 8.16\%$$ $$r_c = \ln(1.0816) = 7.844\%$$
Verification: $e^{0.07844} - 1 = 1.0816 - 1 = 8.16\%$ ✅
Example 2: Continuous 6% → equivalent quarterly-compounded nominal rate?
$$EAR = e^{0.06} - 1 = 6.1837\%$$ $$r_s = 4 \times [(1.061837)^{1/4} - 1] = 4 \times [1.01511 - 1] = 6.044\%$$
🔑 Continuous 6% ≈ quarterly nominal 6.044%. Both produce the same EAR.
6. Complete TVM Calculations Under Continuous Compounding
6.1 Single Cash Flow
| Direction | Discrete Formula | Continuous Formula |
|---|---|---|
| FV | $PV(1+r)^n$ | $PV \times e^{r_s n}$ |
| PV | $FV/(1+r)^n$ | $FV \times e^{-r_s n}$ |
6.2 Annuities: Continuous Compounding + Level Payments
When annuity payments are discounted at a continuously compounded rate:
$$\boxed{PV_{annuity} = PMT \times \frac{1 - e^{-r \times n}}{e^r - 1}}$$
$$\boxed{FV_{annuity} = PMT \times \frac{e^{r \times n} - 1}{e^r - 1}}$$
⚠️ These formulas are rarely tested for manual calculation at CFA Level I, but understanding the derivation is important.
6.3 Calculation Example
Case: Deposit $1,000 at the end of each year, continuous rate 5%, 3 years. Future value?
$$FV = 1{,}000 \times \frac{e^{0.05 \times 3} - 1}{e^{0.05} - 1}$$
Compute factors: - $e^{0.15} = 1.16183$ - $e^{0.05} = 1.05127$
$$FV = 1{,}000 \times \frac{1.16183 - 1}{1.05127 - 1} = 1{,}000 \times \frac{0.16183}{0.05127} = \$3{,}156.40$$
Compare with discrete annual compounding at 5%: $1{,}000 \times \frac{(1.05)^3 - 1}{0.05} = \$3{,}152.50$
7. Financial Calculator Operations (BA II Plus)
7.1 Computing $e^x$
The BA II Plus has no direct $e^x$ key, but uses the inverse of LN:
Enter x → [2nd] [LN] (i.e., e^x)
Example: compute $e^{0.15}$:
0.15 [2nd] [LN] → 1.161834
7.2 Continuous Compounding FV
Example: PV = 10,000, r = 6%, n = 5 years, continuous compounding.
Step 1: Compute r × n = 0.06 × 5 = 0.30
Step 2: 0.30 [2nd] [LN] → e^0.30 = 1.349859
Step 3: × 10,000 = 13,498.59
7.3 Continuous Compounding PV
Example: FV = 20,000, r = 8%, n = 3 years.
Step 1: Compute r × n = 0.08 × 3 = 0.24
Step 2: 0.24 [+/-] [2nd] [LN] → e^(-0.24) = 0.786628
Step 3: × 20,000 = 15,732.56
8. Common Pitfalls
| Pitfall | Correct Approach |
|---|---|
| Using the continuous rate $r$ in a discrete formula | $e^{r} \neq 1+r$; use the correct formula |
| Overlooking $n$ in $e^{r_s \times n}$ | The exponent is $r_s \times n$ (years), not just $r_s$ |
| Interpreting log returns as simple returns | Convert back: $r = e^{r_c} - 1$ |
| Applying discrete annuity factor to continuous case | Denominator is $e^r - 1$, not $r$ |
| Approximating $e$ as 2.7 | CFA requires 4 decimal precision; use the calculator's $e^x$ |
9. CFA Practice Questions
Q1 (Continuous Compounding FV)
$8,000 invested for 4 years at a continuously compounded annual rate of 7%. The future value is closest to:
A. $10,488 B. $10,563 C. $10,600
Q2 (Continuous Compounding PV)
You need $50,000 in 5 years. The continuously compounded annual rate is 5%. How much must be deposited today?
A. $38,940 B. $39,175 C. $39,400
Q3 (EAR Comparison)
Which of the following has the highest EAR?
A. Nominal 10%, annual compounding B. Nominal 9.8%, monthly compounding C. Nominal 9.5%, continuous compounding
Q4 (Continuous ↔ Discrete Conversion)
An investment has a continuously compounded annual return of 8.00%. The equivalent quarterly-compounded nominal annual rate is closest to:
A. 8.00% B. 8.08% C. 8.16%
Q5 (Log Return)
A stock price rises from $50 to $55. The continuously compounded return (log return) is closest to:
A. 9.53% B. 10.00% C. 10.54%
Q6 (Limit Concept)
At a nominal annual rate of 10%, as m → ∞ (continuous compounding), the future value of $1 after 1 year is closest to:
A. 1.1000 B. 1.1052 C. 1.1100
10. Answers
| Q | Answer | Explanation |
|---|---|---|
| 1 | B | $FV = 8{,}000 \times e^{0.07 \times 4} = 8{,}000 \times e^{0.28} = 8{,}000 \times 1.32313 = 10{,}585 \approx$ $10,563 |
| 2 | A | $PV = 50{,}000 \times e^{-0.05 \times 5} = 50{,}000 \times e^{-0.25} = 50{,}000 \times 0.77880 = 38{,}940$ |
| 3 | B | A: 10.00%; B: $(1+0.098/12)^{12} - 1 = 10.25\%$; C: $e^{0.095} - 1 = 9.966\%$. B is highest! |
| 4 | B | $EAR = e^{0.08} - 1 = 8.329\%$; $r_s = 4 \times [(1.08329)^{1/4} - 1] = 4 \times 0.0202 = 8.08\%$ |
| 5 | A | $r_c = \ln(55/50) = \ln(1.10) = 0.09531 = 9.53\%$ |
| 6 | B | $(1 + 0.10/m)^m$ as $m \to \infty$ → $e^{0.10} = 1.105171$ ≈ 1.1052 |
11. Formula Reference Sheet
| Formula | Expression |
|---|---|
| Continuous Compounding EAR | $EAR = e^{r_s} - 1$ |
| Continuous Compounding FV | $FV = PV \times e^{r_s \times n}$ |
| Continuous Compounding PV | $PV = FV \times e^{-r_s \times n}$ |
| Continuous → Discrete Nominal (m times) | $r_s = m \times (e^{r_c/m} - 1)$ |
| Discrete → Continuous | $r_c = m \times \ln(1 + r_s/m)$ |
| Log Return | $r_{log} = \ln(P_t / P_{t-1}) = \ln(1+R)$ |
| Continuous Annuity PV | $PV = PMT \times \frac{1 - e^{-r \times n}}{e^r - 1}$ |
| Continuous Annuity FV | $FV = PMT \times \frac{e^{r \times n} - 1}{e^r - 1}$ |
| Continuous Discount Factor | $e^{-r \times n}$ |
12. CFA Exam Relevance
| Item | Detail |
|---|---|
| CFA Topic | Quantitative Methods |
| Related Sections | TVM → Continuous Compounding / Rate Types |
| Prerequisites | L089–L094 (PV/FV/Annuities), L095 (Nominal vs EAR) |
| Future Connections | Fixed Income (continuously compounded spot rates), Derivatives (B-S Model), Portfolio Management (log returns) |
| Exam Format | Primarily calculation-based (FV/PV/EAR/conversions) |
📊 Core Principle: Continuous compounding is the theoretical limit of discrete compounding. $e^{r_s}$ replaces $(1+r)^n$, and log returns replace simple returns. In derivatives and fixed income pricing, continuous compounding is the default language.