📌 课题：单利与复利 —— 利息计算的两种哲学
一、回顾与引入
L087 我们学习了货币时间价值（TVM）的两个核心概念：现值 PV 与终值 FV，以及连接它们的桥梁——利率。我们默认使用了一个关键假设：利息按复利计算。
但在现实世界中，利息并非只有一种算法。银行存款、债券付息、民间借贷、分期付款……不同场景使用不同的利息计算方式，最终结果天差地别。
本课聚焦两种最基础的利息计算方式： - 单利（Simple Interest） - 复利（Compound Interest）
理解二者的区别，是进入 TVM 计算世界的第一个分水岭。
二、核心概念讲解
2.1 什么是单利？
单利（Simple Interest）：利息只在本金上计算，已产生的利息不再生息。
🔑 单利的本质：利息是「线性」的——每一期的利息金额完全相同。
单利公式：
$$I = P \times r \times t$$
$$FV = P \times (1 + r \times t)$$
其中：
- P = 本金（Principal）
- r = 每期利率（年利率）
- t = 期数（年数）
- I = 总利息
例题 1：单利计算
你存入 10,000 元，年利率 5%，存 3 年，按单利计息。到期本利和是多少？
解析：
| 年份 | 期初本金 | 当年利息 | 累计利息 | 期末本利和 |
|---|---|---|---|---|
| 第 1 年 | 10,000 | 500 | 500 | 10,500 |
| 第 2 年 | 10,000 | 500 | 1,000 | 11,000 |
| 第 3 年 | 10,000 | 500 | 1,500 | 11,500 |
直接代公式：
$$FV = 10{,}000 \times (1 + 0.05 \times 3) = 10{,}000 \times 1.15 = 11{,}500$$
答案：11,500 元
利息每年都是固定的 500 元，三年合计 1,500 元——「利息的算术级数增长」。
2.2 什么是复利？
复利（Compound Interest）：利息在每期末加入本金，下一期利息基于「本金 + 已产生利息」计算。
🔑 复利的本质：利息是「指数」的——「利滚利」，利息也在赚利息。
复利公式：
$$FV = P \times (1 + r)^t$$
$$总利息 = P \times [(1 + r)^t - 1]$$
例题 2：相同条件，复利计算
你存入同样的 10,000 元，年利率 5%，存 3 年，但按复利计息。到期本利和是多少？
解析：
| 年份 | 期初本金 | 当年利息 | 期末本利和 |
|---|---|---|---|
| 第 1 年 | 10,000.00 | 500.00 | 10,500.00 |
| 第 2 年 | 10,500.00 | 525.00 | 11,025.00 |
| 第 3 年 | 11,025.00 | 551.25 | 11,576.25 |
直接代公式：
$$FV = 10{,}000 \times (1 + 0.05)^3 = 10{,}000 \times 1.157625 = 11{,}576.25$$
答案：11,576.25 元
利息逐年递增（500 → 525 → 551.25），三年总利息 1,576.25 元，比单利多赚 76.25 元——这就是「利滚利」的效果。
2.3 单利 vs 复利：可视化对比
| 比较维度 | 单利 | 复利 |
|---|---|---|
| 利息计算基础 | 始终基于原始本金 | 基于本金 + 累计利息 |
| 利息增长模式 | 线性增长（算术级数） | 指数增长（几何级数） |
| 公式 | FV = P(1 + rt) | FV = P(1 + r)^t |
| 短期表现 | 差异很小 | 差异很小 |
| 长期表现 | 落后于复利 | 远超单利 |
| 常见场景 | 短期借贷、部分债券应计利息 | 银行存款、投资回报、大多数金融产品 |
核心结论：
🎯 相同利率、相同期限，复利终值 ≥ 单利终值。时间越长、利率越高，差距越大。
三、深入理解：为什么会差这么多？
3.1 72 法则 —— 快速估算复利翻倍时间
72 法则（Rule of 72）：用 72 除以年化收益率，得到本金翻倍所需的大致年数。
$$翻倍年数 \approx \frac{72}{年利率（\%）}$$
| 年利率 | 72 法则估算 | 精确计算 |
|---|---|---|
| 4% | 72/4 = 18 年 | ln(2)/ln(1.04) ≈ 17.67 年 |
| 6% | 72/6 = 12 年 | 11.90 年 |
| 8% | 72/8 = 9 年 | 9.01 年 |
| 10% | 72/10 ≈ 7.2 年 | 7.27 年 |
| 12% | 72/12 = 6 年 | 6.12 年 |
💡 8%~10% 左右的利率，72 法则最准确。超过 20% 偏差会变大，改用 69.3 法则（更精确）。
对单利而言： - 本金翻倍需要的时间 = 1/r（利率 5% 需要 20 年） - 比复利慢得多！
3.2 长期视角：复利的威力
假设本金 10,000 元，年利率 8%，观察单利 vs 复利在不同年限的差异：
| 年限 | 单利 FV | 复利 FV | 差额 |
|---|---|---|---|
| 1 年 | 10,800 | 10,800 | 0 |
| 5 年 | 14,000 | 14,693 | 693 |
| 10 年 | 18,000 | 21,589 | 3,589 |
| 20 年 | 26,000 | 46,610 | 20,610 |
| 30 年 | 34,000 | 100,627 | 66,627 |
| 40 年 | 42,000 | 217,245 | 175,245 |
🔥 30 年后复利终值是单利的 3 倍，40 年后是 5 倍！ 这就是爱因斯坦口中「世界第八大奇迹」的威力。
四、实际应用场景
4.1 单利的典型应用
- 短期贷款（<1 年）：短期拆借市场通常按单利计算
- 债券应计利息（Accrued Interest）：买卖债券时，买方需补偿卖方持有期间的应计利息，按单利日算
- 商业票据 / 国库券折价：用单利折现
- 某些国家的消费信贷：部分市场法律要求用单利
例题 3：债券应计利息
一张面值 1,000 元、票面利率 6%、每年付息一次的债券。上次付息距今已过 90 天。应计利息是多少？（假设 30/360 计息基础）
解析：
$$应计利息 = 1{,}000 \times 6\% \times \frac{90}{360}$$
$$= 1{,}000 \times 0.06 \times 0.25 = 15$$
答案：15 元
债券应计利息用单利——这是 CFA 固定收益部分的常见考点。
4.2 复利的典型应用
- 储蓄存款：银行定期存款使用复利（按约定频率）
- 投资回报计算：股票、基金、组合收益全部用复利
- 债券到期收益率（YTM）：隐含复利假设
- 企业估值（DCF）：现金流折现全部用复利
- 房贷、消费贷：分期付款用复利（等额本息法）
五、单利率与有效年利率
5.1 名义年利率 vs 有效年利率
当复利频率不是一年一次时（如半年复利、季度复利、月复利），就产生了「名义利率」与「实际利率」的差异。
名义年利率（Stated Annual Rate / r_s）：合同上写的年利率，未考虑复利频率。
有效年利率（Effective Annual Rate / EAR）：考虑了复利频率后，实际一年获得的收益率。
$$EAR = \left(1 + \frac{r_s}{m}\right)^m - 1$$
其中 m 为每年复利次数。
例题 4：有效年利率计算
某银行宣传年利率 12%，分别按以下频率复利，求有效年利率： - (1) 每年复利 - (2) 每半年复利 - (3) 每季度复利 - (4) 每月复利
解析：
(1) 每年复利：$EAR = (1 + 0.12)^1 - 1 = 12\%$
(2) 每半年复利：$EAR = (1 + \frac{0.12}{2})^2 - 1 = 1.06^2 - 1 = 12.36\%$
(3) 每季度复利：$EAR = (1 + \frac{0.12}{4})^4 - 1 = 1.03^4 - 1 = 12.55\%$
(4) 每月复利：$EAR = (1 + \frac{0.12}{12})^{12} - 1 = 1.01^{12} - 1 = 12.68\%$
📈 同样名义利率 12%，复利频率越高，实际收益率越高！
5.2 连续复利 —— 复利的终极形态
当复利频率趋于无穷大（每时每刻都在复利），就是连续复利（Continuous Compounding）：
$$FV = P \times e^{r_s \times t}$$
有效年利率： $$EAR = e^{r_s} - 1$$
例题 5：连续复利
名义年利率 12%，连续复利。10,000 元存 3 年，到期本利和是多少？
解析：
$$FV = 10{,}000 \times e^{0.12 \times 3}$$
$$= 10{,}000 \times e^{0.36}$$
$$= 10{,}000 \times 1.4333$$
$$= 14{,}333$$
对比年复利 FV = 10,000 × 1.12³ = 14,049 → 连续复利多赚了 284 元。
六、单利场景下的折现
虽然大部分金融估值用复利，但在短期（<1 年）场景中常用单利折现。
单利折现公式：
$$PV = \frac{FV}{1 + r \times t}$$
例题 6：短期票据定价
一张面值 100 万的商业票据，90 天后到期，市场年利率 4%（单利）。当前合理价格是多少？
解析：
$$PV = \frac{1{,}000{,}000}{1 + 0.04 \times \frac{90}{360}}$$
$$= \frac{1{,}000{,}000}{1 + 0.01} = \frac{1{,}000{,}000}{1.01}$$
$$= 990{,}099.01$$
答案：约 99.01 万元
七、核心公式速查表
| 公式 | 表达式 |
|---|---|
| 单利终值 | $FV = P(1 + r \times t)$ |
| 单利现值 | $PV = \frac{FV}{1 + r \times t}$ |
| 复利终值 | $FV = P(1 + r)^t$ |
| 复利现值 | $PV = \frac{FV}{(1 + r)^t}$ |
| 有效年利率 | $EAR = \left(1 + \frac{r_s}{m}\right)^m - 1$ |
| 连续复利终值 | $FV = P \times e^{r_s \times t}$ |
| 连续复利 EAR | $EAR = e^{r_s} - 1$ |
| 72 法则 | $翻倍年数 \approx \frac{72}{r(\%)}$ |
八、常见错误与应试提示
⚠️ CFA 考试中容易踩的坑
- 不区分名义利率与有效利率：题目给的是名义利率 + 复利频率时，必须换算为 EAR 或按期利率（r_s/m）计算
- 单利/复利混用：债券 Accrued Interest 一定用单利！YTM 计算一定用复利！
- 复利频率搞错：半年复利 → r/2, n×2；季度复利 → r/4, n×4
- 连续复利公式混淆：FV = P × e^(r×t)，不是 P × (1+r)^t
- EAR 忘了减 1：EAR = (1 + r_s/m)^m − 1，最后那个 −1 经常被漏掉
✅ 应试技巧
- 拿到利率题第一反应：「这个利率是单利还是复利？复利频率是多少？」
- 遇到「等价年化收益率」→ 算 EAR，然后用 EAR 统一比较
- 短期（<1 年）注意题目是否暗示使用单利（如：bank discount yield）
- 计算器（BA II Plus）操作技巧：用 [2nd] [ICONV] 功能快速换算 NOM/EFF
九、本课小结
| 层次 | 要点 |
|---|---|
| 🟢 基础 | 单利 = 利息不生息；复利 = 利滚利 |
| 🟡 进阶 | 复利频率越高，有效年利率越高；连续复利是极限 |
| 🔴 高频考点 | EAR 计算、短期单利折现、债券应计利息（单利） |
| 🧠 思维升级 | 「复利思维」不仅是数学公式，更是长期主义的哲学——坚持做正确的事，让时间成为你的朋友 |
十、练习题
Q1（基础概念）
单利与复利最本质的区别是什么？ - A) 单利用加法，复利用乘法 - B) 单利只在本金上计息，复利在「本金 + 已产生利息」上计息 - C) 单利适用于长期贷款，复利适用于短期贷款 - D) 单利的利率更高
Q2（计算·单利）
本金 50,000 元，年利率 4%，单利，存 5 年。到期本利和是多少？ - A) 52,000 元 - B) 54,000 元 - C) 60,000 元 - D) 60,833 元
Q3（计算·复利）
本金 20,000 元，年利率 6%，复利，存 4 年。到期本利和最接近？ - A) 24,800 元 - B) 25,250 元 - C) 25,249 元 - D) 26,765 元
Q4（计算·有效年利率）
名义年利率 8%，每季度复利一次。有效年利率（EAR）最接近？ - A) 8.00% - B) 8.16% - C) 8.24% - D) 8.33%
Q5（应用·单利折现）
一笔 50 万元的款项将在 180 天后到账，市场年利率 3%（按单利，30/360 计息基础）。该笔款项的现值最接近？ - A) 485,437 元 - B) 492,611 元 - C) 497,512 元 - D) 470,000 元
答案与解析
Q1 答案：B
单利只在本金上计息，复利在本金 + 已产生利息上计息。A 说法模糊（不准确），C 正好说反了，D 利率高低与利息计算方式无关。
Q2 答案：C
FV = 50,000 × (1 + 0.04 × 5) = 50,000 × 1.20 = 60,000 元 每年利息 = 50,000 × 4% = 2,000，5 年共 10,000 利息。
Q3 答案：B
FV = 20,000 × (1 + 0.06)⁴ = 20,000 × 1.262477 ≈ 25,249.54 最接近 25,250 元（选项 B）。注意 C（25,249）也很接近，考试时注意选项精度。
Q4 答案：C
EAR = (1 + 0.08/4)⁴ − 1 = 1.02⁴ − 1 = 1.082432 − 1 = 8.2432% ≈ 8.24%
Q5 答案：B
PV = 500,000 / (1 + 0.03 × 180/360) = 500,000 / (1 + 0.015) = 500,000 / 1.015 ≈ 492,611 元
🧠 下节预告：L089 将进入「复利频率与计息期」——详解不同复利频率下的计算技巧、BA II Plus 计算器的实操，以及名义利率与 EAR 的深层关系。
📌 Topic: Simple vs Compound Interest — Two Philosophies of Interest Calculation
1. Review & Introduction
In L087, we covered the two core concepts of Time Value of Money (TVM): Present Value (PV) and Future Value (FV), along with the bridge that connects them — the interest rate. We assumed by default that interest compounds.
But in the real world, interest isn't calculated in just one way. Bank deposits, bond coupons, private loans, installment payments... different scenarios use different interest calculation methods, and the final results can differ dramatically.
This lesson focuses on the two most fundamental forms of interest: - Simple Interest - Compound Interest
Understanding the difference between the two is the first major watershed in mastering TVM calculations.
2. Core Concepts
2.1 What is Simple Interest?
Simple Interest: Interest is calculated only on the original principal. Interest earned does not earn additional interest.
🔑 The essence of simple interest: Interest grows linearly — every period earns exactly the same amount of interest.
Simple Interest Formulas:
$$I = P \times r \times t$$
$$FV = P \times (1 + r \times t)$$
Where:
- P = Principal
- r = Interest rate per period (annual rate)
- t = Number of periods (years)
- I = Total interest
Example 1: Simple Interest Calculation
You deposit $10,000 at 5% annual interest, simple interest, for 3 years. What is the maturity value?
Solution:
| Year | Beginning Principal | Annual Interest | Cumulative Interest | Ending Balance |
|---|---|---|---|---|
| Year 1 | 10,000 | 500 | 500 | 10,500 |
| Year 2 | 10,000 | 500 | 1,000 | 11,000 |
| Year 3 | 10,000 | 500 | 1,500 | 11,500 |
Using the formula:
$$FV = 10{,}000 \times (1 + 0.05 \times 3) = 10{,}000 \times 1.15 = 11{,}500$$
Answer: $11,500
Interest is a fixed $500 each year, totaling $1,500 over three years — arithmetic growth.
2.2 What is Compound Interest?
Compound Interest: Interest is added to the principal at the end of each period, and the next period's interest is calculated on the new, larger balance.
🔑 The essence of compound interest: Interest grows exponentially — "interest on interest."
Compound Interest Formulas:
$$FV = P \times (1 + r)^t$$
$$Total\ Interest = P \times [(1 + r)^t - 1]$$
Example 2: Same Conditions, Compound Interest
You deposit the same $10,000 at 5% annual interest, but with annual compounding, for 3 years. What is the maturity value?
Solution:
| Year | Beginning Balance | Annual Interest | Ending Balance |
|---|---|---|---|
| Year 1 | 10,000.00 | 500.00 | 10,500.00 |
| Year 2 | 10,500.00 | 525.00 | 11,025.00 |
| Year 3 | 11,025.00 | 551.25 | 11,576.25 |
Using the formula:
$$FV = 10{,}000 \times (1 + 0.05)^3 = 10{,}000 \times 1.157625 = 11{,}576.25$$
Answer: $11,576.25
Interest increases every year (500 → 525 → 551.25). Total interest = $1,576.25 — $76.25 more than simple interest. That's the power of compounding.
2.3 Simple vs Compound: Visual Comparison
| Dimension | Simple Interest | Compound Interest |
|---|---|---|
| Interest basis | Always based on original principal | Based on principal + accumulated interest |
| Growth pattern | Linear (arithmetic) | Exponential (geometric) |
| Formula | FV = P(1 + rt) | FV = P(1 + r)^t |
| Short-term | Minimal difference | Minimal difference |
| Long-term | Falls behind compound | Far exceeds simple |
| Common applications | Short-term loans, bond accrued interest | Bank deposits, investment returns, most financial products |
Key Takeaway:
🎯 For the same rate and same horizon, compound interest FV ≥ simple interest FV. The longer the time and higher the rate, the larger the gap.
3. Deep Dive: Why Such a Big Difference?
3.1 The Rule of 72 — Quick Compounding Estimates
Rule of 72: Divide 72 by the annual rate of return to estimate the approximate number of years needed to double your money.
$$Doubling\ Years \approx \frac{72}{Annual\ Rate\ (\%)}$$
| Annual Rate | Rule of 72 Estimate | Exact Calculation |
|---|---|---|
| 4% | 72/4 = 18 years | ln(2)/ln(1.04) ≈ 17.67 years |
| 6% | 72/6 = 12 years | 11.90 years |
| 8% | 72/8 = 9 years | 9.01 years |
| 10% | 72/10 = 7.2 years | 7.27 years |
| 12% | 72/12 = 6 years | 6.12 years |
💡 The Rule of 72 is most accurate around 8%–10%. For rates above 20%, use the Rule of 69.3 for greater precision.
For simple interest: - Doubling time = 1/r (at 5%, it takes 20 years) - Much slower than compound interest!
3.2 The Long View: The Power of Compounding
Assume $10,000 principal, 8% annual rate. Simple vs compound across different time horizons:
| Years | Simple Interest FV | Compound Interest FV | Difference |
|---|---|---|---|
| 1 | 10,800 | 10,800 | 0 |
| 5 | 14,000 | 14,693 | 693 |
| 10 | 18,000 | 21,589 | 3,589 |
| 20 | 26,000 | 46,610 | 20,610 |
| 30 | 34,000 | 100,627 | 66,627 |
| 40 | 42,000 | 217,245 | 175,245 |
🔥 After 30 years, compound FV is 3× simple FV. After 40 years, it's 5×! This is why Einstein allegedly called compound interest "the eighth wonder of the world."
4. Real-World Applications
4.1 Typical Uses of Simple Interest
- Short-term loans (< 1 year): Money market instruments often use simple interest
- Bond accrued interest: When trading bonds, the buyer compensates the seller for accrued interest using simple interest (day-count basis)
- Commercial paper / T-bill discounting: Uses simple interest discounting
- Some consumer credit markets: Certain jurisdictions mandate simple interest
Example 3: Bond Accrued Interest
A bond with $1,000 face value, 6% coupon rate, pays annual interest. The last coupon payment was 90 days ago. What is the accrued interest? (Assume 30/360 day-count basis.)
Solution:
$$Accrued\ Interest = 1{,}000 \times 6\% \times \frac{90}{360}$$
$$= 1{,}000 \times 0.06 \times 0.25 = 15$$
Answer: $15
Bond accrued interest uses simple interest — a frequently tested concept in CFA Fixed Income.
4.2 Typical Uses of Compound Interest
- Savings deposits: Bank time deposits use compound interest (at stated frequency)
- Investment return calculations: Stocks, funds, portfolio returns all use compound interest
- Bond yield to maturity (YTM): Implicitly assumes compounding
- Corporate valuation (DCF): All cash flow discounting uses compound interest
- Mortgages, consumer loans: Installment payments use compound interest (equal amortization method)
5. Stated Annual Rate vs Effective Annual Rate
5.1 Nominal Rate vs Effective Rate
When compounding frequency is not annual (e.g., semiannual, quarterly, monthly), a gap emerges between the "nominal rate" and the "actual rate."
Stated Annual Rate (r_s): The contractual annual rate, without accounting for compounding frequency.
Effective Annual Rate (EAR): The actual annual return after accounting for compounding frequency.
$$EAR = \left(1 + \frac{r_s}{m}\right)^m - 1$$
Where m is the number of compounding periods per year.
Example 4: EAR Calculation
A bank advertises a 12% annual rate. Calculate the EAR under the following compounding frequencies: - (1) Annual compounding - (2) Semiannual compounding - (3) Quarterly compounding - (4) Monthly compounding
Solution:
(1) Annual: $EAR = (1 + 0.12)^1 - 1 = 12\%$
(2) Semiannual: $EAR = (1 + \frac{0.12}{2})^2 - 1 = 1.06^2 - 1 = 12.36\%$
(3) Quarterly: $EAR = (1 + \frac{0.12}{4})^4 - 1 = 1.03^4 - 1 = 12.55\%$
(4) Monthly: $EAR = (1 + \frac{0.12}{12})^{12} - 1 = 1.01^{12} - 1 = 12.68\%$
📈 Same nominal 12% rate — the higher the compounding frequency, the higher the effective return!
5.2 Continuous Compounding — The Ultimate Form
As compounding frequency approaches infinity (compounding every instant), we get continuous compounding:
$$FV = P \times e^{r_s \times t}$$
Effective Annual Rate: $$EAR = e^{r_s} - 1$$
Example 5: Continuous Compounding
Nominal annual rate 12%, continuous compounding. $10,000 deposited for 3 years. What is the maturity value?
Solution:
$$FV = 10{,}000 \times e^{0.12 \times 3}$$
$$= 10{,}000 \times e^{0.36}$$
$$= 10{,}000 \times 1.4333$$
$$= 14{,}333$$
Compare with annual compounding FV = 10,000 × 1.12³ = 14,049 → continuous compounding earns an extra $284.
6. Discounting with Simple Interest
While most financial valuation uses compound interest, short-term scenarios (<1 year) often use simple interest discounting.
Simple Interest Discounting Formula:
$$PV = \frac{FV}{1 + r \times t}$$
Example 6: Short-Term Instrument Pricing
A commercial paper with $1,000,000 face value matures in 90 days. The market annual rate is 4% (simple interest). What is the fair price today?
Solution:
$$PV = \frac{1{,}000{,}000}{1 + 0.04 \times \frac{90}{360}}$$
$$= \frac{1{,}000{,}000}{1 + 0.01} = \frac{1{,}000{,}000}{1.01}$$
$$= 990{,}099.01$$
Answer: Approximately $990,099
7. Formula Cheat Sheet
| Formula | Expression |
|---|---|
| Simple Interest FV | $FV = P(1 + r \times t)$ |
| Simple Interest PV | $PV = \frac{FV}{1 + r \times t}$ |
| Compound Interest FV | $FV = P(1 + r)^t$ |
| Compound Interest PV | $PV = \frac{FV}{(1 + r)^t}$ |
| Effective Annual Rate | $EAR = \left(1 + \frac{r_s}{m}\right)^m - 1$ |
| Continuous Compounding FV | $FV = P \times e^{r_s \times t}$ |
| Continuous Compounding EAR | $EAR = e^{r_s} - 1$ |
| Rule of 72 | $Doubling\ Years \approx \frac{72}{r(\%)}$ |
8. Common Mistakes & Exam Tips
⚠️ Pitfalls to Avoid on the CFA Exam
- Confusing nominal and effective rates: When the question gives a nominal rate + compounding frequency, you must convert to EAR or use the periodic rate (r_s/m)
- Mixing up simple and compound interest: Bond accrued interest ALWAYS uses simple interest! YTM calculations ALWAYS use compound interest!
- Compounding frequency errors: Semiannual → r/2, n×2; Quarterly → r/4, n×4
- Confusing continuous compounding formula: FV = P × e^(r×t), NOT P × (1+r)^t
- Forgetting the −1 in EAR: EAR = (1 + r_s/m)^m − 1 — that final −1 is often omitted
✅ Exam Tips
- First reaction to any interest rate problem: "Is this simple or compound? What is the compounding frequency?"
- When you see "equivalent annualized yield" → calculate EAR, then compare uniformly using EAR
- For short-term (<1 year), watch for hints that simple interest is expected (e.g., bank discount yield)
- Calculator (BA II Plus) tip: Use [2nd] [ICONV] to quickly convert between NOM and EFF
9. Lesson Summary
| Level | Key Points |
|---|---|
| 🟢 Basic | Simple interest = no interest on interest; Compound interest = interest on interest |
| 🟡 Intermediate | Higher compounding frequency → higher EAR; continuous compounding is the limit |
| 🔴 High-Yield | EAR calculation, short-term simple discounting, bond accrued interest (simple) |
| 🧠 Mindset | "Compound thinking" isn't just math — it's a philosophy of long-termism: keep doing the right things, and let time be your ally |
10. Practice Questions
Q1 (Basic Concept)
What is the most fundamental difference between simple and compound interest? - A) Simple interest uses addition, compound interest uses multiplication - B) Simple interest calculates on principal only; compound interest calculates on principal + accumulated interest - C) Simple interest applies to long-term loans; compound interest applies to short-term loans - D) Simple interest has a higher rate
Q2 (Calculation · Simple Interest)
Principal = $50,000, annual rate = 4%, simple interest, 5 years. What is the maturity value? - A) $52,000 - B) $54,000 - C) $60,000 - D) $60,833
Q3 (Calculation · Compound Interest)
Principal = $20,000, annual rate = 6%, compound interest, 4 years. Maturity value is closest to: - A) $24,800 - B) $25,250 - C) $25,249 - D) $26,765
Q4 (Calculation · EAR)
Nominal annual rate = 8%, compounded quarterly. The Effective Annual Rate (EAR) is closest to: - A) 8.00% - B) 8.16% - C) 8.24% - D) 8.33%
Q5 (Application · Simple Discounting)
A $500,000 payment will be received in 180 days. The market annual rate is 3% (simple interest, 30/360 day-count basis). The present value is closest to: - A) $485,437 - B) $492,611 - C) $497,512 - D) $470,000
Answers & Explanations
Q1 Answer: B
Simple interest calculates on principal only; compound interest calculates on principal + accumulated interest. Option A is vague and imprecise; C reverses the typical application; D is irrelevant — the rate itself is independent of the interest calculation method.
Q2 Answer: C
FV = 50,000 × (1 + 0.04 × 5) = 50,000 × 1.20 = $60,000 Annual interest = $50,000 × 4% = $2,000; 5 years total = $10,000 interest.
Q3 Answer: B
FV = 20,000 × (1 + 0.06)⁴ = 20,000 × 1.262477 ≈ $25,249.54 Closest to $25,250 (Option B). Note that C ($25,249) is very close — pay attention to rounding conventions on the exam.
Q4 Answer: C
EAR = (1 + 0.08/4)⁴ − 1 = 1.02⁴ − 1 = 1.082432 − 1 = 8.2432% ≈ 8.24%
Q5 Answer: B
PV = 500,000 / (1 + 0.03 × 180/360) = 500,000 / (1 + 0.015) = 500,000 / 1.015 ≈ $492,611
🧠 Next Up: L089 — "Compounding Frequency & Periodicity" — detailed calculation techniques for different compounding frequencies, BA II Plus calculator walkthrough, and the deeper relationship between nominal rates and EAR.