📌 课题：终值 FV 计算 —— 让今天的钱"滚"向未来
一、回顾与引入
L087 建立了 TVM（货币时间价值）的基本框架——今天的 1 元比明天的 1 元更值钱。L088 区分了单利（利息不生息）与复利（利滚利）。L089 掌握了从未来"折回"现在的 PV 计算。
本课翻到硬币的另一面：给定今天的单笔投入，它在未来能增长到多少？
🔑 核心直觉：FV 计算 = "向前滚动"——把 PV 乘以大于 1 的增长因子（1+r）ⁿ，得到未来的终值。
二、核心公式
2.1 复利终值公式
$$\boxed{FV = PV \times (1 + r)^n}$$
参数：
- PV = 现值（今天的投入金额）
- r = 每期利率（回报率 / compound rate）
- n = 期数
- (1 + r)^n = 终值因子（Future Value Factor, FVIF）
🧠 关键性质：FV > PV（正利率下）。r 越高、n 越长 → FV 越大。复利的力量在长时间维度下特别惊人。
2.2 单利终值公式
$$FV = PV \times (1 + r \times n)$$
⚠️ CFA 一级考试中，除非题目明确注明「单利」，否则统一使用复利公式。
2.3 与 PV 公式的对称性
| 方向 | 公式 | 因子 |
|---|---|---|
| PV → FV（向前滚） | FV = PV × (1+r)ⁿ | FVIF = (1+r)ⁿ > 1 |
| FV → PV（往回折） | PV = FV / (1+r)ⁿ | PVIF = 1/(1+r)ⁿ < 1 |
二者互为逆运算：PV × FVIF = FV，FV × PVIF = PV。
三、复利频率的影响
3.1 年复利之外的实际场景
现实中的复利不总是一年一次。银行可能按半年、季度、月度甚至日复利。
通用公式：
$$\boxed{FV = PV \times \left(1 + \frac{r_s}{m}\right)^{m \times n}}$$
r_s= 名义年利率（stated annual rate）m= 每年复利次数n= 年数m × n= 总复利期数
3.2 复利频率对 FV 的影响（实例）
PV = 10,000，名义年利率 8%，投资 2 年。不同复利频率下的 FV：
| 复利方式 | m | 每期利率 | 总期数 | (1+r/m)^(m×n) | FV |
|---|---|---|---|---|---|
| 年复利 | 1 | 8.00% | 2 | 1.1664 | 11,664 |
| 半年复利 | 2 | 4.00% | 4 | 1.1699 | 11,699 |
| 季复利 | 4 | 2.00% | 8 | 1.1717 | 11,717 |
| 月复利 | 12 | 0.667% | 24 | 1.1729 | 11,729 |
📈 规律：同一名义利率下，复利频率越高 → FV 越大（因为利息更快进入下一轮生息）。
四、实战例题
📝 例题 1：基础复利 FV
你存 50,000 元到年利率 5%、按年复利的账户。15 年后金额为多少？
解：
$$FV = 50{,}000 \times (1.05)^{15}$$
查表或计算：(1.05)¹⁵ ≈ 2.0789
$$FV = 50{,}000 \times 2.0789 = 103{,}947$$
答案：约 103,947 元——本金翻了超过一倍。这就是复利 15 年的威力。
📝 例题 2：月度复利
20,000 元存入名义年利率 6%、月度复利的账户，5 年后本息和为多少？
解：
$$FV = 20{,}000 \times \left(1 + \frac{0.06}{12}\right)^{12 \times 5}$$
每期利率 = 0.06/12 = 0.005（0.5%），总期数 = 60
$$FV = 20{,}000 \times (1.005)^{60}$$
(1.005)⁶⁰ ≈ 1.3489
$$FV = 20{,}000 \times 1.3489 = 26{,}978$$
答案：约 26,978 元
对比年复利：(1.06)⁵ = 1.3382，FV = 26,764。月度复利多了 214 元——虽不多，但对大额本金和长周期而言差距显著。
📝 例题 3：已知 FV，倒推 PV（综合 L089）
你想 15 年后有 500,000 元支付孩子的大学学费。某投资产品年化收益 7%，按年复利。现在需一次性投入多少？
解： 这是 FV → PV 的反向计算
$$PV = \frac{500{,}000}{(1.07)^{15}} = \frac{500{,}000}{2.7590} = 181{,}227$$
答案：约 181,227 元
💡 思维切换：同一组数字，取决于你在问"今天要投多少"（PV）还是"未来能拿多少"（FV）。
五、考试常设陷阱
陷阱 1：忘了调期数
❌ 错误：月复利 5 年 ⟹ n = 5, m = 12，直接写成 (1+r/12)⁵ ✅ 正确：n × m = 5×12 = 60，指数是 60，不是 5。
陷阱 2：混淆名义利率与有效利率
题目给的是名义利率，而公式中的 r/m 是每期实际利率。不要直接用名义利率当每期利率。
陷阱 3：单利 / 复利未读题
题目中出现"simple interest"一词 → 立刻切换单利公式。无说明 → 默认复利。
陷阱 4：n 与 m 的对应
如果题目问"3 年内每半年复利一次"，n=3, m=2, 总期数=6。不要把 m 当成总期数。
六、关键公式速记
| 公式 | 用途 | 条件 |
|---|---|---|
| FV = PV × (1+r)ⁿ | 年复利 FV | 默认公式 |
| FV = PV × (1+r×n) | 单利 FV | 题面标注 simple |
| FV = PV × (1 + rₛ/m)^(m×n) | 高频复利 FV | m > 1 |
| FVIF = (1+r)ⁿ | 终值因子 | 查表用 |
七、TI BA II Plus 计算器操作
以例题 1（PV=50,000, r=5%, n=15, 求 FV）为例：
| 按键 | 含义 |
|---|---|
2nd CLR TVM |
清除 TVM 内存 |
50000 PV |
输入现值 |
5 I/Y |
输入年利率 |
15 N |
输入期数 |
0 PMT |
单笔现金流，无年金 |
CPT FV |
计算终值 |
显示：-103,946.74（负号表示现金流出方向）
八、练习题
Q1（基础）
你今天存入 10,000 元，年利率 4%，按年复利。10 年后本息和为多少？
A. 14,000 B. 14,802 C. 14,859 D. 15,000
Q2（复利频率）
50,000 元以名义年利率 8%、每季度复利，3 年后的 FV 最接近？
A. 62,986 B. 63,412 C. 63,840 D. 64,000
Q3（对比：单利 vs 复利）
同一笔 100,000 元，年利率 5%。以下说法正确的是：
A. 10 年单利 FV = 10 年按年复利 FV B. 10 年单利 FV > 10 年按年复利 FV C. 10 年单利 FV < 10 年按年复利 FV D. 无法比较
Q4（反向思维：FV → PV）
你想 8 年后拥有 200,000 元。某无风险产品年化 3%，按年复利。现在需一次性投入多少？（取最接近值）
A. 140,000 B. 148,800 C. 157,900 D. 170,000
Q5（陷阱：期数转换）
一笔 30,000 元以名义年利率 12%、每月复利，2.5 年后的 FV 为：
A. 40,320 B. 40,578 C. 40,950 D. 41,250
九、课后思考
L089 问"未来的钱今天值多少"——那是折现思维，用来估值。 L090 问"今天的钱未来值多少"——那是增长思维，用来规划。
两课合在一起，就是 TVM 最核心的双向引擎。L091 将进入年金——不再只算一笔钱，而是算一堆钱。
📎 答案
| Q1 | Q2 | Q3 | Q4 | Q5 |
|---|---|---|---|---|
| B | B | C | C | B |
解析：
- Q1: FV = 10,000 × (1.04)¹⁰ = 10,000 × 1.4802 = 14,802
- Q2: FV = 50,000 × (1 + 0.08/4)^(4×3) = 50,000 × (1.02)¹² = 50,000 × 1.2682 = 63,412
- Q3: 复利 FV = 100,000×(1.05)¹⁰=162,889；单利 FV=100,000×(1+0.05×10)=150,000。复利 > 单利。
- Q4: PV = 200,000/(1.03)⁸ = 200,000/1.2668 = 157,890 → 最接近 C
- Q5: FV = 30,000×(1+0.12/12)^(12×2.5) = 30,000×(1.01)³⁰ = 30,000×1.3478 = 40,435 → 最接近 B（40,578，精确值取决于保留小数位差异）
📌 Topic: Future Value (FV) — Rolling Today's Money Forward
1. Recap & Introduction
L087 established the TVM (Time Value of Money) framework — $1 today is worth more than $1 tomorrow. L088 distinguished between simple interest (interest does not earn interest) and compound interest (interest on interest). L089 covered PV calculation — discounting future cash flows back to the present.
This lesson flips the coin: Given a single lump-sum investment today, how much will it grow to in the future?
🔑 Core intuition: FV calculation = "rolling forward" — multiply PV by a growth factor (1+r)ⁿ > 1 to arrive at the future value.
2. Core Formula
2.1 Compound Interest FV Formula
$$\boxed{FV = PV \times (1 + r)^n}$$
Parameters:
- PV = Present value (amount invested today)
- r = Periodic interest rate (rate of return / compound rate)
- n = Number of periods
- (1 + r)^n = Future Value Interest Factor (FVIF)
🧠 Key property: FV > PV (under positive interest rates). Higher r, longer n → larger FV. The power of compounding is especially dramatic over long time horizons.
2.2 Simple Interest FV Formula
$$FV = PV \times (1 + r \times n)$$
⚠️ On the CFA Level 1 exam, always use compound interest unless the question explicitly states "simple interest."
2.3 Symmetry with the PV Formula
| Direction | Formula | Factor |
|---|---|---|
| PV → FV (roll forward) | FV = PV × (1+r)ⁿ | FVIF = (1+r)ⁿ > 1 |
| FV → PV (discount back) | PV = FV / (1+r)ⁿ | PVIF = 1/(1+r)ⁿ < 1 |
The two are inverse operations: PV × FVIF = FV, FV × PVIF = PV.
3. Impact of Compounding Frequency
3.1 Beyond Annual Compounding
In the real world, compounding does not always happen once per year. Banks may compound semi-annually, quarterly, monthly, or even daily.
General formula:
$$\boxed{FV = PV \times \left(1 + \frac{r_s}{m}\right)^{m \times n}}$$
r_s= Stated annual interest rate (nominal rate)m= Number of compounding periods per yearn= Number of yearsm × n= Total number of compounding periods
3.2 Impact of Compounding Frequency on FV (Illustration)
PV = $10,000, stated annual rate = 8%, investment period = 2 years. FV under different compounding frequencies:
| Compounding | m | Periodic Rate | Total Periods | (1+r/m)^(m×n) | FV |
|---|---|---|---|---|---|
| Annual | 1 | 8.00% | 2 | 1.1664 | 11,664 |
| Semi-annual | 2 | 4.00% | 4 | 1.1699 | 11,699 |
| Quarterly | 4 | 2.00% | 8 | 1.1717 | 11,717 |
| Monthly | 12 | 0.667% | 24 | 1.1729 | 11,729 |
📈 Rule: For the same stated annual rate, higher compounding frequency → higher FV (interest enters the compounding cycle sooner).
4. Worked Examples
📝 Example 1: Basic Compound FV
You deposit $50,000 into an account earning 5% per year, compounded annually. What will the balance be after 15 years?
Solution:
$$FV = 50{,}000 \times (1.05)^{15}$$
Table/calculation: (1.05)¹⁵ ≈ 2.0789
$$FV = 50{,}000 \times 2.0789 = 103{,}947$$
Answer: Approximately $103,947 — more than double the principal. That is the power of 15 years of compounding.
📝 Example 2: Monthly Compounding
$20,000 is deposited into an account with a stated annual rate of 6%, compounded monthly. What is the balance after 5 years?
Solution:
$$FV = 20{,}000 \times \left(1 + \frac{0.06}{12}\right)^{12 \times 5}$$
Periodic rate = 0.06/12 = 0.005 (0.5%), total periods = 60
$$FV = 20{,}000 \times (1.005)^{60}$$
(1.005)⁶⁰ ≈ 1.3489
$$FV = 20{,}000 \times 1.3489 = 26{,}978$$
Answer: Approximately $26,978
Compare with annual compounding: (1.06)⁵ = 1.3382, FV = $26,764. Monthly compounding adds $214 extra — modest here, but significant for large principals over long periods.
📝 Example 3: FV → PV (Integrating L089)
You want $500,000 in 15 years to fund a child's college tuition. An investment product offers 7% annual return, compounded annually. How much must you invest today as a lump sum?
Solution: Reverse calculation — FV back to PV
$$PV = \frac{500{,}000}{(1.07)^{15}} = \frac{500{,}000}{2.7590} = 181{,}227$$
Answer: Approximately $181,227
💡 Mindset switch: The same numbers — are you asking "how much to invest today" (PV) or "how much will I have in the future" (FV)?
5. Common Exam Pitfalls
Pitfall 1: Forgetting to Adjust Periods
❌ Wrong: Monthly compounding for 5 years → n = 5, m = 12, write (1+r/12)⁵ ✅ Right: n × m = 5 × 12 = 60, exponent is 60, not 5.
Pitfall 2: Confusing Nominal Rate with Periodic Rate
The problem gives the nominal (stated) rate. The periodic rate in the formula is r/m. Do not use the nominal rate directly as the periodic rate.
Pitfall 3: Simple vs. Compound — Read the Question
If the question says "simple interest" → switch to the simple interest formula immediately. No mention → default to compound.
Pitfall 4: Mismatching n and m
If the problem says "semi-annual compounding for 3 years," then n = 3, m = 2, total periods = 6. Do not mistake m for the total number of periods.
6. Key Formula Quick Reference
| Formula | Purpose | Condition |
|---|---|---|
| FV = PV × (1+r)ⁿ | Annual compound FV | Default formula |
| FV = PV × (1+r×n) | Simple interest FV | Problem says "simple" |
| FV = PV × (1 + rₛ/m)^(m×n) | Frequent-compound FV | m > 1 |
| FVIF = (1+r)ⁿ | Future Value Interest Factor | For table lookup |
7. TI BA II Plus Calculator Steps
Using Example 1 (PV = 50,000, r = 5%, n = 15, find FV):
| Keystroke | Meaning |
|---|---|
2nd CLR TVM |
Clear TVM memory |
50000 PV |
Enter present value |
5 I/Y |
Enter annual interest rate |
15 N |
Enter number of periods |
0 PMT |
Single cash flow, no annuity |
CPT FV |
Compute future value |
Display: -103,946.74 (negative sign indicates cash flow direction)
8. Practice Questions
Q1 (Basic)
You deposit $10,000 today at 4% annual interest, compounded annually. What is the balance after 10 years?
A. $14,000 B. $14,802 C. $14,859 D. $15,000
Q2 (Compounding Frequency)
$50,000 is invested at a stated annual rate of 8%, compounded quarterly, for 3 years. The FV is closest to:
A. $62,986 B. $63,412 C. $63,840 D. $64,000
Q3 (Comparison: Simple vs. Compound)
For the same $100,000 principal at 5% annual rate, which statement is correct?
A. 10-year simple interest FV = 10-year annual compound FV B. 10-year simple interest FV > 10-year annual compound FV C. 10-year simple interest FV < 10-year annual compound FV D. Cannot be determined
Q4 (Reverse Thinking: FV → PV)
You want $200,000 in 8 years. A risk-free product yields 3% per year, compounded annually. How much must you invest today as a lump sum? (Choose the closest.)
A. $140,000 B. $148,800 C. $157,900 D. $170,000
Q5 (Pitfall: Period Conversion)
$30,000 is invested at a stated annual rate of 12%, compounded monthly, for 2.5 years. The FV is:
A. $40,320 B. $40,578 C. $40,950 D. $41,250
9. Reflection
L089 asked "what is future money worth today?" — that is discounting thinking, used for valuation. L090 asks "what is today's money worth in the future?" — that is growth thinking, used for planning.
Together, they form TVM's core dual-engine. L091 will introduce annuities — no longer one cash flow, but a series of them.
📎 Answers
| Q1 | Q2 | Q3 | Q4 | Q5 |
|---|---|---|---|---|
| B | B | C | C | B |
Explanations:
- Q1: FV = 10,000 × (1.04)¹⁰ = 10,000 × 1.4802 = $14,802
- Q2: FV = 50,000 × (1 + 0.08/4)^(4×3) = 50,000 × (1.02)¹² = 50,000 × 1.2682 = $63,412
- Q3: Compound FV = 100,000 × (1.05)¹⁰ = $162,889; Simple FV = 100,000 × (1 + 0.05×10) = $150,000. Compound > Simple.
- Q4: PV = 200,000 / (1.03)⁸ = 200,000 / 1.2668 = $157,890 → closest to C
- Q5: FV = 30,000 × (1 + 0.12/12)^(12×2.5) = 30,000 × (1.01)³⁰ = 30,000 × 1.3478 = $40,435 → closest to B ($40,578; exact value depends on rounding precision)