📌 课题：用 10 道题检验 L087-L096 全部 TVM 知识掌握程度
一、考试说明
| 项目 | 详情 |
|---|---|
| 总题数 | 10 题 |
| 限时 | 18 分钟（每题约 100 秒） |
| 覆盖范围 | L087-L096 全部知识点 |
| 题型 | 单选题（4 选 1） |
| 分值 | 每题 10 分，满分 100 分 |
| 难度 | CFA 一级真题水平 |
考查知识点分布
| 题号 | 知识点 | 对应课次 |
|---|---|---|
| 1 | PV/FV 基本概念 | L087 |
| 2 | 单利 vs 复利 | L088 |
| 3 | 单笔现金流 PV | L089 |
| 4 | 单笔现金流 FV | L090 |
| 5 | 普通年金 vs 预付年金 | L091 |
| 6 | 年金 PV 计算 | L092 |
| 7 | 永续年金 | L093 |
| 8 | 不均匀现金流 PV | L094 |
| 9 | 名义利率 vs EAR | L095 |
| 10 | 连续复利 | L096 |
二、10 道测试题
题 1（L087 · PV/FV 基本概念）
关于货币时间价值，以下哪项说法是错误的？
A. 在其他条件不变时，利率越高，FV 越高 B. 在其他条件不变时，期数越多，PV 越低 C. 一笔现金流，只要利率为正，其 FV 一定大于等于 PV D. 折现率越高，未来收到的 1 元钱今天值越多的钱
题 2（L088 · 单利 vs 复利）
$10,000 存入银行，名义年利率 6%，期限 3 年。单利与年复利获得的利息差额最接近：
A. $80 B. $110 C. $140 D. $170
题 3（L089 · 单笔现金流 PV）
某公司 7 年后有一笔 $500,000 的债券到期兑付，当前市场上同等风险的折现率为 6.5%（年复利）。这笔未来现金流的现值最接近：
A. $310,500 B. $321,800 C. $330,400 D. $345,200
题 4（L090 · 单笔现金流 FV）
你有一笔 $18,000 的闲置资金，投资于年复利 8.5% 的理财产品，持有 12 年。到期时的终值最接近：
A. $45,600 B. $47,900 C. $49,300 D. $50,200
题 5（L091 · 年金类型识别）
以下关于普通年金与预付年金的描述，哪项是正确的？
A. 普通年金每期期初发生，预付年金每期期末发生 B. 普通年金的 PV 一定大于同等条件下预付年金的 PV C. 预付年金的 FV = 普通年金 FV × (1 + r) D. 普通年金的 FV = 预付年金 FV × (1 + r)
题 6（L092 · 年金 PV 计算）
你计划退休后每年末取出 $45,000 作为生活费，预计退休后生活 25 年。如果退休时的折现率为 5%（年复利），退休那一天你需要准备多少资金？
A. $586,200 B. $634,200 C. $648,800 D. $675,000
题 7（L093 · 永续年金）
某优先股每年固定派息 $6.00，市场上类似风险的优先股收益率为 8%。该优先股的内在价值为：
A. $65.00 B. $72.00 C. $75.00 D. $80.00
题 8（L094 · 不均匀现金流）
一个投资项目未来 4 年的预期现金流为：第 1 年末 $2,000、第 2 年末 $4,000、第 3 年末 $6,000、第 4 年末 $8,000。如果折现率为 8%，该现金流的现值最接近：
A. $15,700 B. $15,940 C. $16,210 D. $16,450
题 9（L095 · 名义利率 vs EAR）
某银行信用卡的名义年利率为 18%，按日复利（一年按 365 天计）。该信用卡的有效年利率（EAR）最接近：
A. 18.36% B. 19.56% C. 19.72% D. 20.12%
题 10（L096 · 连续复利）
$25,000 以连续复利年利率 7% 投资 5 年。终值计算公式为：
A. FV = 25,000 × (1 + 0.07)^5 B. FV = 25,000 × e^(0.07 × 5) C. FV = 25,000 × (1 + 0.07 / 12)^60 D. FV = 25,000 × ln(0.07 × 5)
三、答案与解析
题 1 答案：D
解析：
- A ✅ 正确：FV = PV × (1 + r)^n，r 越大，FV 越大
- B ✅ 正确：PV = FV / (1 + r)^n，n 越大，分母越大，PV 越小
- C ✅ 正确：r > 0 时，(1 + r)^n > 1（n ≥ 1），FV > PV；若 n = 0 则 FV = PV
- D ❌ 错误：折现率越高，PV = FV / (1 + r)^n 越小，未来收到的 1 元今天值更少的钱
常考辨析：分清「今天→未来」用乘法、「未来→今天」用除法。
题 2 答案：B（$110）
计算过程：
- 单利利息 = PV × r × n = 10,000 × 0.06 × 3 = $1,800
- 复利终值 = 10,000 × (1.06)^3 = 10,000 × 1.191016 = $11,910.16
- 复利利息 = 11,910.16 − 10,000 = $1,910.16
- 差额 = 1,910.16 − 1,800 = $110.16 ≈ $110
复利额外收益 = 利滚利（利息再投资产生的利息）。PV 越大、r 越高、n 越长，差额越大。
题 3 答案：B（$321,800）
计算过程：
PV = FV / (1 + r)^n = 500,000 / (1.065)^7
(1.065)^7 = 1.065^3 × 1.065^4 ≈ 1.2079 × 1.2865 = 1.55399
PV = 500,000 / 1.55399 = $321,754 ≈ $321,800
7 年的折现，$500,000 被利率「吃掉」约 36%，只值今天的 $32 万。
题 4 答案：B（$47,900）
计算过程：
FV = PV × (1 + r)^n = 18,000 × (1.085)^12
(1.085)^4 ≈ 1.38586 → (1.085)^12 = (1.38586)^3 ≈ 2.66153
FV = 18,000 × 2.66153 = $47,907.5 ≈ $47,900
8.5% 下 12 年翻到 2.66 倍，展示了复利的威力。
题 5 答案：C
解析：
- A ❌：普通年金 = 期末，预付年金 = 期初（说反了）
- B ❌：预付年金每笔现金流少折现一期 → PV_预付 > PV_普通
- C ✅：预付年金每笔现金流多复利一期 → FV_预付 = FV_普通 × (1 + r)
- D ❌：方向反了
口诀：预付年金「快一步」——更早收钱 → PV 更高，更早投资 → FV 更高。
题 6 答案：B（$634,200）
计算过程：
普通年金 PV = PMT × PVIFA(5%, 25)
PVIFA = [1 − 1/(1.05)^25] / 0.05 1.05^25 ≈ 3.38635 PVIFA = (1 − 1/3.38635) / 0.05 = (1 − 0.29530) / 0.05 = 0.70470 / 0.05 = 14.0940
PV = 45,000 × 14.0940 = $634,230 ≈ $634,200
退休规划是 CFA 一级高频考点：PV_年金 = 退休那一刻需要的总资金。
题 7 答案：C（$75.00）
计算过程：
永续年金 PV = PMT / r = 6.00 / 0.08 = $75.00
优先股是最经典的永续年金：固定股息 ÷ 要求回报率 = 内在价值。
题 8 答案：B（$15,940）
计算过程：
逐笔折现并求和：
| 年份 | 现金流 | 折现因子(1.08^n) | PV |
|---|---|---|---|
| 1 | $2,000 | 1.0800 | $1,851.85 |
| 2 | $4,000 | 1.1664 | $3,429.36 |
| 3 | $6,000 | 1.2597 | $4,763.04 |
| 4 | $8,000 | 1.3605 | $5,880.24 |
| 合计 | $15,924.49 |
最接近选项 B $15,940。
不均匀现金流没有简便公式，必须逐笔折现求和。考试时用计算器 CF 工作表快速求解。
题 9 答案：C（19.72%）
计算过程：
EAR = (1 + r/m)^m − 1 = (1 + 0.18/365)^365 − 1
(1 + 0.18/365)^365 = (1.00049315)^365 ≈ e^(0.18 − 0.18^2/(2×365) + ...) ≈ e^0.1798 ≈ 1.1972
EAR = 1.1972 − 1 = 0.1972 = 19.72%
日复利的 EAR（19.72%）显著高于名义利率 18%。复利频率越高 → EAR 越大，但收敛于 e^r − 1（连续复利上限 19.72%）。
题 10 答案：B
解析：
连续复利公式：FV = PV × e^(r × t)
- A：普通年复利公式，非连续复利
- B ✅：连续复利公式，e^(0.07×5) = e^0.35
- C：月复利公式
- D：ln 是对数函数，不是终值计算公式
连续复利 = 复利频率的极限（m → ∞）。公式是 FV = PV × e^(r×n)，是 CFA 一级必记公式之一。
四、成绩评估
| 正确题数 | 分数 | 等级 | 建议 |
|---|---|---|---|
| 9-10 | 90-100 | ⭐ 优秀 | 直接进入下一个模块 |
| 7-8 | 70-89 | ✅ 良好 | 错题重点复习，可继续推进 |
| 5-6 | 50-69 | ⚠️ 待加强 | 重做对应课次的练习，尤其是年金和 EAR |
| 0-4 | 0-49 | 🔴 需重修 | 建议重看 L087-L096 核心课次再测 |
🎯 TVM 模块结业提示： L098 是 TVM 模块的最后一次课。通过本次测试即完成 CFA 一级定量方法中最核心的时间价值模块。下一模块将进入描述性统计与概率基础。
Topic: 10 Questions to Assess Your Mastery of All TVM Concepts from L087-L096
I. Test Overview
| Item | Detail |
|---|---|
| Total Questions | 10 |
| Time Limit | 18 minutes (~100 seconds per question) |
| Coverage | All topics from L087-L096 |
| Format | Single-choice (4 options per question) |
| Scoring | 10 points each, 100 total |
| Difficulty | CFA Level I exam level |
Topic Distribution
| Q# | Topic | Lesson |
|---|---|---|
| 1 | PV/FV Core Concepts | L087 |
| 2 | Simple vs Compound Interest | L088 |
| 3 | Single Cash Flow PV | L089 |
| 4 | Single Cash Flow FV | L090 |
| 5 | Ordinary Annuity vs Annuity Due | L091 |
| 6 | Annuity PV Calculation | L092 |
| 7 | Perpetuity | L093 |
| 8 | Uneven Cash Flow PV | L094 |
| 9 | Nominal Rate vs EAR | L095 |
| 10 | Continuous Compounding | L096 |
II. 10 Test Questions
Question 1 (L087 · PV/FV Core Concepts)
Regarding the time value of money, which of the following statements is incorrect?
A. Holding other factors constant, the higher the interest rate, the higher the FV B. Holding other factors constant, the more periods, the lower the PV C. As long as the interest rate is positive, the FV of a cash flow is always greater than or equal to its PV D. The higher the discount rate, the more a future dollar is worth today
Question 2 (L088 · Simple vs Compound Interest)
$10,000 is deposited in a bank account with a nominal annual rate of 6% for 3 years. The difference in interest earned between simple interest and annual compounding is closest to:
A. $80 B. $110 C. $140 D. $170
Question 3 (L089 · Single Cash Flow PV)
A company has a $500,000 bond maturing in 7 years. The current market discount rate for similar risk is 6.5% (annual compounding). The present value of this future cash flow is closest to:
A. $310,500 B. $321,800 C. $330,400 D. $345,200
Question 4 (L090 · Single Cash Flow FV)
You have $18,000 in idle funds and invest it at 8.5% annual compounding for 12 years. The future value at maturity is closest to:
A. $45,600 B. $47,900 C. $49,300 D. $50,200
Question 5 (L091 · Annuity Type Identification)
Which of the following statements about ordinary annuities and annuities due is correct?
A. Ordinary annuity payments occur at the beginning of each period; annuity due payments occur at the end B. The PV of an ordinary annuity is always greater than the PV of an equivalent annuity due C. FV of annuity due = FV of ordinary annuity × (1 + r) D. FV of ordinary annuity = FV of annuity due × (1 + r)
Question 6 (L092 · Annuity PV Calculation)
You plan to withdraw $45,000 at the end of each year during retirement, which you expect to last 25 years. If the discount rate at retirement is 5% (annual compounding), how much capital do you need on the day you retire?
A. $586,200 B. $634,200 C. $648,800 D. $675,000
Question 7 (L093 · Perpetuity)
A preferred stock pays a fixed annual dividend of $6.00. The market yield on similar-risk preferred stocks is 8%. The intrinsic value of this preferred stock is:
A. $65.00 B. $72.00 C. $75.00 D. $80.00
Question 8 (L094 · Uneven Cash Flows)
An investment project has the following expected cash flows: Year 1: $2,000, Year 2: $4,000, Year 3: $6,000, Year 4: $8,000 (all at year-end). If the discount rate is 8%, the present value of these cash flows is closest to:
A. $15,700 B. $15,940 C. $16,210 D. $16,450
Question 9 (L095 · Nominal Rate vs EAR)
A credit card charges a nominal annual rate of 18% compounded daily (365 days per year). The effective annual rate (EAR) on this card is closest to:
A. 18.36% B. 19.56% C. 19.72% D. 20.12%
Question 10 (L096 · Continuous Compounding)
$25,000 is invested at 7% continuously compounded annual rate for 5 years. The correct formula for the future value is:
A. FV = 25,000 × (1 + 0.07)^5 B. FV = 25,000 × e^(0.07 × 5) C. FV = 25,000 × (1 + 0.07 / 12)^60 D. FV = 25,000 × ln(0.07 × 5)
III. Answers and Explanations
Question 1 – Answer: D
Explanation:
- A ✓ Correct: FV = PV × (1 + r)^n; as r increases, FV increases
- B ✓ Correct: PV = FV / (1 + r)^n; as n increases, the denominator grows, so PV decreases
- C ✓ Correct: When r > 0, (1 + r)^n > 1 (for n ≥ 1), so FV > PV; if n = 0 then FV = PV
- D ✗ Incorrect: The higher the discount rate, the lower the PV (PV = FV / (1 + r)^n). A future dollar is worth less today when discount rates are higher.
Key distinction: Today → Future uses multiplication; Future → Today uses division.
Question 2 – Answer: B ($110)
Calculation:
- Simple interest = PV × r × n = 10,000 × 0.06 × 3 = $1,800
- Compound FV = 10,000 × (1.06)^3 = 10,000 × 1.191016 = $11,910.16
- Compound interest = 11,910.16 − 10,000 = $1,910.16
- Difference = 1,910.16 − 1,800 = $110.16 ≈ $110
The extra return from compounding = interest on interest. Larger PV, higher r, and longer n all increase the gap.
Question 3 – Answer: B ($321,800)
Calculation:
PV = FV / (1 + r)^n = 500,000 / (1.065)^7
(1.065)^7 = 1.065^3 × 1.065^4 ≈ 1.2079 × 1.2865 = 1.55399
PV = 500,000 / 1.55399 = $321,754 ≈ $321,800
After 7 years of discounting at 6.5%, $500,000 is worth only about $322k today — nearly 36% lost to the time value of money.
Question 4 – Answer: B ($47,900)
Calculation:
FV = PV × (1 + r)^n = 18,000 × (1.085)^12
(1.085)^4 ≈ 1.38586 → (1.085)^12 = (1.38586)^3 ≈ 2.66153
FV = 18,000 × 2.66153 = $47,907.5 ≈ $47,900
At 8.5%, $18,000 grows to 2.66× over 12 years — the power of compounding in action.
Question 5 – Answer: C
Explanation:
- A ✗: Ordinary annuity = end of period; Annuity due = beginning of period (reversed)
- B ✗: Annuity due payments are discounted one fewer period → PV_due > PV_ordinary
- C ✓: Each annuity due payment compounds one extra period → FV_due = FV_ordinary × (1 + r)
- D ✗: Direction is reversed
Memory aid: Annuity due is always "one step ahead" — higher PV (money received sooner) and higher FV (money invested sooner).
Question 6 – Answer: B ($634,200)
Calculation:
Ordinary annuity PV = PMT × PVIFA(5%, 25)
PVIFA = [1 − 1/(1.05)^25] / 0.05 1.05^25 ≈ 3.38635 PVIFA = (1 − 1/3.38635) / 0.05 = (1 − 0.29530) / 0.05 = 0.70470 / 0.05 = 14.0940
PV = 45,000 × 14.0940 = $634,230 ≈ $634,200
Retirement planning is a high-frequency CFA Level I topic. The PV of this annuity = total capital needed at retirement.
Question 7 – Answer: C ($75.00)
Calculation:
Perpetuity PV = PMT / r = 6.00 / 0.08 = $75.00
Preferred stock is the classic perpetuity application: fixed dividend ÷ required return = intrinsic value.
Question 8 – Answer: B ($15,940)
Calculation:
Discount each cash flow and sum:
| Year | Cash Flow | Discount Factor (1.08^n) | PV |
|---|---|---|---|
| 1 | $2,000 | 1.0800 | $1,851.85 |
| 2 | $4,000 | 1.1664 | $3,429.36 |
| 3 | $6,000 | 1.2597 | $4,763.04 |
| 4 | $8,000 | 1.3605 | $5,880.24 |
| Total | $15,924.49 |
Closest to option B $15,940.
Uneven cash flows have no shortcut formula — must discount each individually. On the exam, use the CF worksheet on your calculator.
Question 9 – Answer: C (19.72%)
Calculation:
EAR = (1 + r/m)^m − 1 = (1 + 0.18/365)^365 − 1
(1 + 0.18/365)^365 ≈ e^0.1798 ≈ 1.1972
EAR = 1.1972 − 1 = 0.1972 = 19.72%
Daily compounding EAR (19.72%) is significantly higher than the nominal 18%. Higher compounding frequency → higher EAR, converging to e^r − 1 under continuous compounding.
Question 10 – Answer: B
Explanation:
Continuous compounding formula: FV = PV × e^(r × t)
- A: Standard annual compounding — not continuous
- B ✓: Correct continuous compounding formula: e^(0.07 × 5) = e^0.35
- C: Monthly compounding formula
- D: ln is the natural logarithm, not a future value formula
Continuous compounding = the limit as compounding frequency approaches infinity. FV = PV × e^(r×n) is one of the must-memorize formulas for CFA Level I.
IV. Score Assessment
| Correct Answers | Score | Grade | Recommendation |
|---|---|---|---|
| 9–10 | 90–100 | ⭐ Excellent | Move directly to next module |
| 7–8 | 70–89 | ✅ Good | Review incorrect questions, then proceed |
| 5–6 | 50–69 | ⚠️ Needs Work | Re-do exercises for missed topics, especially annuities and EAR |
| 0–4 | 0–49 | 🔴 Retake Recommended | Review core lessons L087–L096 before retesting |
🎯 TVM Module Complete: L098 is the final lesson of the TVM module. Passing this test means you have mastered the most critical quantitative topic in CFA Level I. Next module: Descriptive Statistics & Probability Basics.