📌 课题：把 L087-L096 学到的所有 TVM 工具，放到 20 道实战题里检验一遍
一、为什么需要这节综合练习？
TVM（Time Value of Money）是 CFA 一级定量方法中最核心的计算模块，涵盖：
| 课次 | 知识点 |
|---|---|
| L087 | TVM 导论：PV、FV 概念 |
| L088 | 单利 vs 复利 |
| L089 | PV 计算（单笔现金流） |
| L090 | FV 计算（单笔现金流） |
| L091 | 年金类型：普通年金 vs 预付年金 |
| L092 | 年金 PV/FV 计算 |
| L093 | 永续年金 |
| L094 | 不均匀现金流 PV 计算 |
| L095 | 名义利率 vs 有效年利率（EAR） |
| L096 | 连续复利 |
💡 CFA 考试中，TVM 相关题目占定量方法 30-40%，且经常与其他模块（固定收益、股票、公司金融）交叉出题。计算速度直接决定做题节奏。
二、练习策略
- 先做分类题：按知识块逐个击破
- 再做综合题：模拟真实考试的混搭出题
- 计时：每题限时 90 秒，20 题共计 30 分钟
- 卡住就跳：不纠结，最后集中看解析
三、分类练习（12 题 · 按知识点分组）
A 组：单笔现金流 PV/FV（3 题）
题 A1（PV 基础） 你想在 8 年后有 $200,000 用作孩子大学教育金。如果年复利 6%，现在需要一次性存入多少？
A. $118,000 B. $125,480 C. $132,600
题 A2（FV 基础） $15,000 投资于年复利 7.5% 的账户，持有 10 年。终值最接近多少？
A. $28,900 B. $30,900 C. $31,800
题 A3（单利 vs 复利） $10,000 存 5 年，名义年利率 8%。单利 vs 年复利的利息差额为：
A. $0 B. $332 C. $693
B 组：年金 PV/FV（3 题）
题 B1（普通年金 FV） 每月末存入 $500，年利率 6% 按月复利，连续存 5 年。终值最接近：
A. $33,000 B. $34,885 C. $36,000
题 B2（预付年金 PV） 你中了彩票：每年初领取 $30,000，共领 10 年。如果折现率 5%，这笔年金现值是多少？
A. $231,630 B. $243,210 C. $255,360
题 B3（年金 PV → PMT） 你借了一笔 $250,000 房贷，年利率 4.8% 按月复利，30 年等额本息还清。每月还款额是多少？(月还款发生在月末)
A. $1,280 B. $1,312 C. $1,350
C 组：永续年金（2 题）
题 C1（永续年金基础） 一只优先股每年派息 $4.50，要求回报率 7%。该优先股的内在价值是多少？
A. $60.00 B. $64.29 C. $70.00
题 C2（增长永续年金） 某股票明年预计派息 $2.00，此后股利以每年 3% 永续增长。如果要求回报率 10%，该股票现值为：
A. $20.00 B. $28.57 C. $33.33
D 组：名义利率 vs EAR + 连续复利（4 题）
题 D1（EAR 比较） 以下哪个 EAR 最高？
A. 名义 12%，季复利 B. 名义 11.8%，月复利 C. 名义 11.5%，连续复利
题 D2（连续复利 FV） $20,000 以连续复利年利率 6.5% 投资 5 年。终值最接近：
A. $27,200 B. $27,680 C. $28,010
题 D3（连续复利 PV） 6 年后需要 $40,000，连续复利年利率 5%，现在应存入多少？
A. $29,630 B. $29,850 C. $30,120
题 D4（对数收益率） 股票从 $80 涨到 $92，连续复利收益率（对数收益率）最接近：
A. 13.98% B. 15.00% C. 15.50%
四、综合题（8 题 · 混合考点）
题 5（不均匀现金流） 一个项目未来 3 年的现金流分别为：第 1 年末 $3,000、第 2 年末 $5,000、第 3 年末 $8,000。如果折现率 9%，该现金流的 PV 是多少？
A. $12,500 B. $12,780 C. $13,150
题 6（两步折现） 某投资：第 3 年末开始每年末领取 $4,000，共领 5 年（即第 3、4、5、6、7 年末各有 $4,000）。折现率 8%，该现金流在今天的现值是多少？
A. $12,760 B. $13,680 C. $14,220
题 7（EAR + 年金组合） 名义年利率 8% 按季复利。每月末存入 $300，连存 3 年，终值是多少？
A. $12,000 B. $12,180 C. $12,360
题 8（还款方式比较） 借款 $100,000，名义年利率 6% 按年复利，10 年还清。
- 方案 1：等额本息，每年末还款
- 方案 2：等额本金，每年末还本金 $10,000 + 剩余本金 × 6%
方案 1 的总利息比方案 2 多多少？
A. $3,180 B. $2,580 C. 方案 2 利息更多
题 9（连续复利 + 年金混合） 每年末存 $5,000，连续复利年利率 4%，存 5 年。终值最接近：
A. $26,500 B. $27,100 C. $27,500
题 10（退休规划） 你 35 岁，计划 60 岁退休。退休后预计活到 85 岁，每年需要 $80,000（年末提取）。退休前和退休期间，投资年回报率均为 7%。
问：从今天开始到 60 岁，每年末需要存多少钱？
A. $10,200 B. $12,600 C. $14,800
题 11（实际利率） 名义年利率 9%，通货膨胀率 3%，根据 Fisher 方程，实际利率最接近：
A. 5.50% B. 5.83% C. 6.00%
题 12（永续 + 延迟） 某捐赠基金计划：从第 5 年末开始，每年末永久支付 $10,000。折现率 6%。该捐赠的现值是：
A. $125,000 B. $133,333 C. $166,667
五、答案与解析
A 组
| 题号 | 答案 | 解析 |
|---|---|---|
| A1 | B | $PV = \frac{200{,}000}{(1.06)^8} = \frac{200{,}000}{1.59385} = 125{,}482$ → $125,480 |
| A2 | B | $FV = 15{,}000 \times (1.075)^{10} = 15{,}000 \times 2.06103 = 30{,}915$ → $30,900 |
| A3 | B | 复利：$10{,}000 \times (1.08)^5 = 14{,}693$；单利：$10{,}000 + 5 \times 800 = 14{,}000$；差 = $693$ → $693 ⚠️ 注意选项。$14,693 - 14,000 = 693$，选 C |
🔑 A3 更正：差额 = $14,693 - $14,000 = $693 → 选 C
B 组
| 题号 | 答案 | 解析 |
|---|---|---|
| B1 | B | N=60, I/Y=6/12=0.5, PMT=500, PV=0 → FV = $34,885 ✅ |
| B2 | B | 预付年金：BGN 模式。N=10, I/Y=5, PMT=30,000, FV=0 → PV = $243,210。验证：普通年金 PV = $231,630 → × 1.05 = $243,211 ✅ |
| B3 | B | N=360, I/Y=4.8/12=0.4, PV=250,000, FV=0 → PMT = $1,311.78 → $1,312 |
C 组
| 题号 | 答案 | 解析 |
|---|---|---|
| C1 | B | $PV = \frac{4.50}{0.07} = 64.29$ ✅ |
| C2 | B | $PV = \frac{2.00}{0.10 - 0.03} = \frac{2.00}{0.07} = 28.57$ ✅ |
D 组
| 题号 | 答案 | 解析 |
|---|---|---|
| D1 | B | A: $(1.03)^4 - 1 = 12.551\%$；B: $(1+0.118/12)^{12} - 1 = 12.467\%$；C: $e^{0.115} - 1 = 12.188\%$ → A 最大！ ⚠️ 重算：B = $(1.00983)^{12} - 1 = 12.468\%$ → A 胜出，选 A |
| D2 | B | $FV = 20{,}000 \times e^{0.065 \times 5} = 20{,}000 \times e^{0.325} = 20{,}000 \times 1.38403 = 27{,}681$ → $27,680 |
| D3 | A | $PV = 40{,}000 \times e^{-0.05 \times 6} = 40{,}000 \times e^{-0.30} = 40{,}000 \times 0.74082 = 29{,}633$ → $29,630 |
| D4 | A | $r_c = \ln(92/80) = \ln(1.15) = 0.13976 = 13.98\%$ ✅ |
综合题
| 题号 | 答案 | 解析 |
|---|---|---|
| 5 | B | $PV = \frac{3{,}000}{1.09} + \frac{5{,}000}{1.09^2} + \frac{8{,}000}{1.09^3} = 2{,}752 + 4{,}210 + 6{,}177 = 13{,}139$ → $13,150（选 C） ⚠️ 精确：$2{,}752.29 + 4{,}209.84 + 6{,}177.37 = 13{,}139.50$ → 最接近 C $13,150 |
| 6 | A | 两步法：① 第 2 年末年金 PV = PMT × PVIFA(8%,5) = $4{,}000 × 3.9927 = $15,971；② 再折现 2 年 = $15,971 / 1.08² = $15,971 / 1.1664 = $13,693。→ $13,680（选 B） |
🔑 题 6 更正：第 3 年末开始领取 → 普通年金在第 2 年末的 PV = $15,971，折现到 t=0 → $15,971/(1.08²) = $13,693 → 选 B $13,680
| 题号 | 答案 | 解析 |
|---|---|---|
| 7 | B | 先求月 EAR 等价：$(1+0.08/4)^{4} = (1+r_{monthly})^{12}$ → $1.08243^{1/12} - 1 = 0.00662$；N=36, I/Y=0.662, PMT=300 → FV ≈ $12,180 ✅ |
| 8 | B | 方案 1：PMT = $100{,}000 / PVIFA(6%,10) = $100{,}000 / 7.3601 = $13,587；总利息 = 13,587 × 10 − 100,000 = $35,870。方案 2：利息 = 6% × (100,000+90,000+…+10,000) = 6% × 550,000 = $33,000。差额 = 35,870 − 33,000 = $2,870。最接近 B $2,580 |
🔑 题 8 精确计算：方案 1 年金因子 = $(1-1.06^{-10})/0.06 = 7.36009$ → PMT = $13,586.80；总偿还 = $135,868；利息 = $35,868。方案 2：利息 = 0.06 × (100k+90k+...+10k) = 0.06 × 550,000 = $33,000。差额 = $2,868 → 最接近 B $2,580（选项精度有限）
| 题号 | 答案 | 解析 |
|---|---|---|
| 9 | B | $FV = 5{,}000 \times \frac{e^{0.04 \times 5} - 1}{e^{0.04} - 1} = 5{,}000 \times \frac{1.22140 - 1}{1.04081 - 1} = 5{,}000 \times \frac{0.22140}{0.04081} = 5{,}000 \times 5.4249 = 27{,}125$ → $27,100 ✅ |
| 10 | B | ① 退休时需 PV（60 岁时）：$PMT=80{,}000, N=25, I/Y=7 → PV = 80{,}000 × 11.6536 = $932,288。② 储蓄期 25 年（35→60）：FV=$932,288, N=25, I/Y=7 → PMT = $932,288 / 63.249 = $14,740。→ C $14,800 ✅ ⚠️ 重算：FVIFA(7%,25) = (1.07²⁵−1)/0.07 = (5.4274−1)/0.07 = 63.2488；PMT = 932,288 / 63.2488 = 14,741 → 选 C |
🔑 题 10 更正：退休后 PV @ 60 = $80,000 × PVIFA(7%,25) = $80,000 × 11.6536 = $932,288。年储蓄 PMT = $932,288 / FVIFA(7%,25) = $932,288 / 63.249 = $14,740 → C
| 题号 | 答案 | 解析 |
|---|---|---|
| 11 | B | Fisher: $(1+r_{nom}) = (1+r_{real})(1+\pi)$ → $1.09 = (1+r_{real})(1.03)$ → $r_{real} = 1.09/1.03 - 1 = 5.825\%$ → 5.83% |
| 12 | A | 永续年金 PV @ 第 4 年末 = $10,000/0.06 = $166,667。折现到 t=0：$166,667/(1.06⁴) = $166,667/1.26248 = $132,012。→ $132,013 不在选项中！⚠️ 检查：PV @ t=4 还是 t=5？第 5 年末开始支付 → 永续年金 PV = $10,000/0.06 = $166,667 位于 第 4 年末（因为永续年金 PV 公式给出的是第一笔支付前一期）。折现 4 年 → $166,667/(1.06)⁴ = 132,013。但选项只有 $125,000 / $133,333 / $166,667。$166,667/1.06⁴ ≈ $132,013，最接近 $133,333 → 选 B $133,333 |
六、正确率评估
| 正确数 | 等级 | 下一步 |
|---|---|---|
| 18-20 | 优秀 ✅ | TVM 已扎实，直接进入 L098 周测 |
| 14-17 | 良好 👍 | 回顾错题对应的知识点（表格见下） |
| 10-13 | 需要加强 ⚠️ | 重做对应课次的练习题 |
| <10 | 建议回炉 🔄 | 从 L087 重新过一遍，尤其是年金和 EAR |
七、错题 → 知识点对照表
| 错题号 | 知识点 | 回看课次 |
|---|---|---|
| A1-A3 | 单笔 PV/FV、单利复利 | L088/L089/L090 |
| B1 | 普通年金终值 | L091/L092 |
| B2 | 预付年金现值 | L091/L092 |
| B3 | 年金 PMT 计算 | L092 |
| C1 | 永续年金 | L093 |
| C2 | 增长永续年金 | L093 |
| D1 | EAR 比较 | L095 |
| D2-D3 | 连续复利 FV/PV | L096 |
| D4 | 对数收益率 | L096 |
| 5 | 不均匀现金流 PV | L094 |
| 6 | 延迟年金 PV（两步折现） | L092/L094 |
| 7 | EAR + 年金组合 | L092/L095 |
| 8 | 还款方式比较 | L092 |
| 9 | 连续复利 + 年金 | L096 |
| 10 | 退休规划（多阶段） | L089/L092 |
| 11 | Fisher 实际利率 | L095 |
| 12 | 延迟永续年金 | L093 |
八、金融计算器速查
基础设置
| 操作 | 按键 |
|---|---|
| 清空 TVM | [2nd] [FV]（CLR TVM） |
| 设为 BGN 模式 | [2nd] [PMT] [2nd] [ENTER]（显示 BGN） |
| 退出 BGN 模式 | [2nd] [PMT] [2nd] [ENTER]（BGN 消失） |
| 设 P/Y=C/Y | [2nd] [I/Y] → 输入 → [ENTER] |
TVM 五键快速口诀
| 已知 | 求 | 口诀 |
|---|---|---|
| N, I/Y, PV, PMT | FV | 输入 4 个 → [CPT] [FV] |
| N, I/Y, PMT, FV | PV | 输入 4 个 → [CPT] [PV] |
| N, I/Y, PV, FV | PMT | PV 和 FV 符号相反（一进一出） |
| PV, PMT, FV | N | 计算期数，注意单位匹配 |
| N, PV, PMT, FV | I/Y | 注意结果是期间利率 |
常用函数
| 运算 | 按键 |
|---|---|
| 计算 eˣ | x [2nd] [LN] |
| 计算 ln(x) | x [LN] |
| 计算 yˣ | y [yˣ] x [=] |
| 倒数 | x [1/x] |
| 符号切换 | [+/-] |
年金快捷判断
| 情况 | 模式 | 判断标志 |
|---|---|---|
| 每月末 | END | "月末/年末/期末" → 普通年金 |
| 每月初 | BGN | "月初/年初/期初/立即" → 预付年金 |
| 先存后取 | END | 常规储蓄 |
| 先付后用 | BGN | 房租/租赁/彩票领取 |
九、核心公式速查（全 TVM 模块）
单笔现金流
| 公式 | 表达式 |
|---|---|
| FV（离散） | $FV = PV(1 + r)^n$ |
| PV（离散） | $PV = FV / (1 + r)^n$ |
| FV（连续） | $FV = PV \times e^{r \times n}$ |
| PV（连续） | $PV = FV \times e^{-r \times n}$ |
| 单利 | $FV = PV(1 + r \times n)$ |
年金
| 公式 | 表达式 |
|---|---|
| 普通年金 PV | $PV = PMT \times \frac{1 - (1+r)^{-n}}{r}$ |
| 普通年金 FV | $FV = PMT \times \frac{(1+r)^n - 1}{r}$ |
| 预付年金 PV | $PV = PMT \times \frac{1 - (1+r)^{-n}}{r} \times (1+r)$ |
| 预付年金 FV | $FV = PMT \times \frac{(1+r)^n - 1}{r} \times (1+r)$ |
永续年金
| 公式 | 表达式 |
|---|---|
| 普通永续 | $PV = PMT / r$ |
| 增长永续 | $PV = PMT_1 / (r - g)$ |
| 延迟永续 | $PV = (PMT / r) / (1+r)^t$ |
利率换算
| 公式 | 表达式 |
|---|---|
| 期间利率 | $r_{period} = r_s / m$ |
| EAR（m 次复利） | $EAR = (1 + r_s/m)^m - 1$ |
| EAR（连续） | $EAR = e^{r_s} - 1$ |
| Fisher 实际利率 | $(1+r_{nom}) = (1+r_{real})(1+\pi)$ |
| 对数收益率 | $r_{log} = \ln(P_t / P_{t-1})$ |
十、下一课预告
L098 TVM 周测 — 10 道定量题，模拟 CFA 真实考试的 TVM 部分节奏。建议在完成本课练习且正确率 ≥ 70% 后再进入。
📊 核心信条：TVM 是 CFA 一级的计算基石。PV、FV、年金、EAR、连续复利 → 五把钥匙，一把不开错门。速度 + 准确 = 考场上的从容。
📌 Topic: Test all TVM tools from L087–L096 in 20 real exam-style problems
一、Why This Comprehensive Practice?
TVM (Time Value of Money) is the core calculation module in CFA Level 1 Quantitative Methods:
| Lesson | Topic |
|---|---|
| L087 | TVM Introduction: PV and FV Concepts |
| L088 | Simple Interest vs Compound Interest |
| L089 | PV Calculation (Single Cash Flow) |
| L090 | FV Calculation (Single Cash Flow) |
| L091 | Annuity Types: Ordinary Annuity vs Annuity Due |
| L092 | Annuity PV/FV Calculation |
| L093 | Perpetuity |
| L094 | Uneven Cash Flow PV Calculation |
| L095 | Nominal Rate vs Effective Annual Rate (EAR) |
| L096 | Continuous Compounding |
💡 In the CFA exam, TVM-related questions account for 30–40% of Quantitative Methods and frequently cross over into other modules (Fixed Income, Equity, Corporate Finance). Calculation speed directly determines your exam pacing.
二、Practice Strategy
- Categorized problems first: Tackle each knowledge block one at a time
- Mixed problems second: Simulate real exam-style mixed questions
- Time yourself: 90 seconds per question, 30 minutes total for 20 questions
- Skip if stuck: Don't dwell — review explanations at the end
三、Categorized Practice (12 Problems · By Topic)
Group A: Single Cash Flow PV/FV (3 problems)
Problem A1 (PV Basics) You want $200,000 in 8 years for your child's college fund. If the annual compound rate is 6%, how much must you deposit today as a lump sum?
A. $118,000 B. $125,480 C. $132,600
Problem A2 (FV Basics) $15,000 is invested in an account earning 7.5% compounded annually for 10 years. The future value is closest to:
A. $28,900 B. $30,900 C. $31,800
Problem A3 (Simple vs Compound) $10,000 is deposited for 5 years at a nominal annual rate of 8%. The difference in interest earned between simple interest and annual compounding is:
A. $0 B. $332 C. $693
Group B: Annuity PV/FV (3 problems)
Problem B1 (Ordinary Annuity FV) $500 is deposited at the end of each month. The annual interest rate is 6% compounded monthly. Deposits continue for 5 years. The future value is closest to:
A. $33,000 B. $34,885 C. $36,000
Problem B2 (Annuity Due PV) You win a lottery: receive $30,000 at the beginning of each year for 10 years. If the discount rate is 5%, the present value of this annuity is:
A. $231,630 B. $243,210 C. $255,360
Problem B3 (Annuity PV → PMT) You borrow $250,000 as a mortgage. The annual rate is 4.8% compounded monthly, to be repaid over 30 years with equal month-end payments. The monthly payment is closest to:
A. $1,280 B. $1,312 C. $1,350
Group C: Perpetuity (2 problems)
Problem C1 (Basic Perpetuity) A preferred stock pays an annual dividend of $4.50. The required rate of return is 7%. The intrinsic value of this preferred stock is:
A. $60.00 B. $64.29 C. $70.00
Problem C2 (Growing Perpetuity) A stock is expected to pay a dividend of $2.00 next year, with dividends growing at 3% per year in perpetuity thereafter. If the required return is 10%, the present value of the stock is:
A. $20.00 B. $28.57 C. $33.33
Group D: Nominal Rate vs EAR + Continuous Compounding (4 problems)
Problem D1 (EAR Comparison) Which of the following has the highest EAR?
A. Nominal 12%, compounded quarterly B. Nominal 11.8%, compounded monthly C. Nominal 11.5%, compounded continuously
Problem D2 (Continuous Compounding FV) $20,000 is invested at a continuously compounded annual rate of 6.5% for 5 years. The future value is closest to:
A. $27,200 B. $27,680 C. $28,010
Problem D3 (Continuous Compounding PV) You need $40,000 in 6 years. The continuously compounded annual rate is 5%. How much should you deposit today?
A. $29,630 B. $29,850 C. $30,120
Problem D4 (Log Return) A stock rises from $80 to $92. The continuously compounded rate of return (log return) is closest to:
A. 13.98% B. 15.00% C. 15.50%
四、Mixed Problems (8 Problems · Cross-Topic)
Problem 5 (Uneven Cash Flows) A project generates the following cash flows: $3,000 at end of Year 1, $5,000 at end of Year 2, $8,000 at end of Year 3. With a discount rate of 9%, the PV of these cash flows is closest to:
A. $12,500 B. $12,780 C. $13,150
Problem 6 (Two-Step Discounting) An investment pays $4,000 at the end of each year for 5 years, starting at the end of Year 3 (i.e., payments at end of Years 3, 4, 5, 6, and 7). With a discount rate of 8%, the present value today is closest to:
A. $12,760 B. $13,680 C. $14,220
Problem 7 (EAR + Annuity Combo) The nominal annual rate is 8% compounded quarterly. $300 is deposited at the end of each month for 3 years. The future value is closest to:
A. $12,000 B. $12,180 C. $12,360
Problem 8 (Repayment Method Comparison) Borrow $100,000 at a nominal annual rate of 6% compounded annually, to be repaid over 10 years.
- Plan 1: Equal annual payments at year-end (fully amortizing)
- Plan 2: Equal principal repayment: $10,000 principal + interest on remaining balance at each year-end
The total interest under Plan 1 exceeds Plan 2 by approximately:
A. $3,180 B. $2,580 C. Plan 2 has more interest
Problem 9 (Continuous Compounding + Annuity) $5,000 is deposited at the end of each year. The continuously compounded annual rate is 4%. Deposits continue for 5 years. The future value is closest to:
A. $26,500 B. $27,100 C. $27,500
Problem 10 (Retirement Planning) You are 35 years old and plan to retire at 60. You expect to live until 85 and need $80,000 per year (withdrawn at year-end) during retirement. The investment return is 7% per year both before and during retirement.
How much must you save at the end of each year from now until age 60?
A. $10,200 B. $12,600 C. $14,800
Problem 11 (Real Interest Rate) The nominal annual rate is 9% and the inflation rate is 3%. According to the Fisher equation, the real interest rate is closest to:
A. 5.50% B. 5.83% C. 6.00%
Problem 12 (Delayed Perpetuity) An endowment fund plans to distribute $10,000 at the end of each year in perpetuity, starting at the end of Year 5. With a discount rate of 6%, the present value of this endowment is closest to:
A. $125,000 B. $133,333 C. $166,667
五、Answers and Explanations
Group A
| # | Answer | Explanation |
|---|---|---|
| A1 | B | $PV = \frac{200{,}000}{(1.06)^8} = \frac{200{,}000}{1.59385} = 125{,}482 \approx \$125{,}480$ |
| A2 | B | $FV = 15{,}000 \times (1.075)^{10} = 15{,}000 \times 2.06103 = 30{,}915 \approx \$30{,}900$ |
| A3 | C | Compound: $10{,}000 \times (1.08)^5 = 14{,}693$; Simple: $10{,}000 + 5 \times 800 = 14{,}000$; Difference = $693 |
Group B
| # | Answer | Explanation |
|---|---|---|
| B1 | B | N=60, I/Y=6/12=0.5, PMT=500, PV=0 → FV = $34,885 |
| B2 | B | Annuity Due (BGN mode): N=10, I/Y=5, PMT=30,000, FV=0 → PV = $243,210. Check: Ordinary annuity PV = $231,630 × 1.05 = $243,211 ✓ |
| B3 | B | N=360, I/Y=4.8/12=0.4, PV=250,000, FV=0 → PMT = $1,311.78 ≈ $1,312 |
Group C
| # | Answer | Explanation |
|---|---|---|
| C1 | B | $PV = \frac{4.50}{0.07} = 64.29$ |
| C2 | B | $PV = \frac{2.00}{0.10 - 0.03} = \frac{2.00}{0.07} = 28.57$ |
Group D
| # | Answer | Explanation |
|---|---|---|
| D1 | A | A: $(1.03)^4 - 1 = 12.551\%$; B: $(1+0.118/12)^{12} - 1 = 12.468\%$; C: $e^{0.115} - 1 = 12.188\%$ → A is highest |
| D2 | B | $FV = 20{,}000 \times e^{0.065 \times 5} = 20{,}000 \times e^{0.325} = 20{,}000 \times 1.38403 = 27{,}681 \approx \$27{,}680$ |
| D3 | A | $PV = 40{,}000 \times e^{-0.05 \times 6} = 40{,}000 \times e^{-0.30} = 40{,}000 \times 0.74082 = 29{,}633 \approx \$29{,}630$ |
| D4 | A | $r_c = \ln(92/80) = \ln(1.15) = 0.13976 = 13.98\%$ |
Mixed Problems
| # | Answer | Explanation |
|---|---|---|
| 5 | C | $PV = \frac{3{,}000}{1.09} + \frac{5{,}000}{1.09^2} + \frac{8{,}000}{1.09^3} = 2{,}752.29 + 4{,}209.84 + 6{,}177.37 = 13{,}139.50 \approx \$13{,}150$ |
| 6 | B | Two-step: ① Annuity PV at t=2: $4{,}000 × PVIFA(8\%,5) = 4{,}000 × 3.9927 = 15{,}970.80$; ② Discount to t=0: $15{,}970.80 / 1.08^2 = 13{,}693.41 \approx \$13{,}680$ |
| 7 | B | First find equivalent monthly rate: $(1+0.08/4)^4 = (1+r_{monthly})^{12}$ → $r_{monthly} = 1.08243^{1/12} - 1 = 0.00662$; N=36, I/Y=0.662, PMT=300 → FV ≈ $12,180 |
| 8 | B | Plan 1: PMT = 100,000 / PVIFA(6\%,10) = 100,000 / 7.3601 = $13,587; Total interest = 135,870 − 100,000 = $35,870. Plan 2: Interest = 6\% × (100k+90k+…+10k) = 6\% × 550,000 = $33,000. Difference = $35,870 − $33,000 = $2,870 ≈ $2,580 |
| 9 | B | $FV = 5{,}000 \times \frac{e^{0.20} - 1}{e^{0.04} - 1} = 5{,}000 \times \frac{0.22140}{0.04081} = 5{,}000 \times 5.4249 = 27{,}125 \approx \$27{,}100$ |
| 10 | C | Step ①: PV needed at age 60 = 80,000 × PVIFA(7\%,25) = 80,000 × 11.6536 = 932,288. Step ②: Annual savings = 932,288 / FVIFA(7\%,25) = 932,288 / 63.249 = $14,740 ≈ $14,800 |
| 11 | B | Fisher: $(1+r_{nom}) = (1+r_{real})(1+\pi)$ → $1.09 = (1+r_{real})(1.03)$ → $r_{real} = 1.09/1.03 - 1 = 5.825\% \approx 5.83\%$ |
| 12 | B | Perpetuity PV at end of Year 4: $10{,}000 / 0.06 = 166{,}667$. Discount 4 years to t=0: $166{,}667 / 1.06^4 = 132{,}013$. Closest option is $133,333 → B |
六、Score Evaluation
| Correct | Grade | Next Step |
|---|---|---|
| 18–20 | Excellent ✅ | TVM is solid — proceed directly to L098 Weekly Quiz |
| 14–17 | Good 👍 | Review the topics corresponding to incorrect answers (see table below) |
| 10–13 | Needs Improvement ⚠️ | Redo practice problems from the corresponding lessons |
| <10 | Recommended Retake 🔄 | Revisit L087–L096, especially annuities and EAR |
七、Errors → Topic Mapping
| Wrong Problem | Topic | Review Lesson |
|---|---|---|
| A1–A3 | Single Cash Flow PV/FV, Simple vs Compound | L088/L089/L090 |
| B1 | Ordinary Annuity FV | L091/L092 |
| B2 | Annuity Due PV | L091/L092 |
| B3 | Annuity PMT Calculation | L092 |
| C1 | Perpetuity | L093 |
| C2 | Growing Perpetuity | L093 |
| D1 | EAR Comparison | L095 |
| D2–D3 | Continuous Compounding FV/PV | L096 |
| D4 | Log Return | L096 |
| 5 | Uneven Cash Flow PV | L094 |
| 6 | Delayed Annuity PV (Two-Step) | L092/L094 |
| 7 | EAR + Annuity Combo | L092/L095 |
| 8 | Repayment Method Comparison | L092 |
| 9 | Continuous Compounding + Annuity | L096 |
| 10 | Retirement Planning (Multi-Stage) | L089/L092 |
| 11 | Fisher Real Interest Rate | L095 |
| 12 | Delayed Perpetuity | L093 |
八、Financial Calculator Quick Reference
Basic Settings
| Operation | Keys |
|---|---|
| Clear TVM | [2nd] [FV] (CLR TVM) |
| Set BGN Mode | [2nd] [PMT] [2nd] [ENTER] (BGN displayed) |
| Exit BGN Mode | [2nd] [PMT] [2nd] [ENTER] (BGN disappears) |
| Set P/Y = C/Y | [2nd] [I/Y] → enter value → [ENTER] |
TVM Five-Key Quick Guide
| Known | Solve For | Note |
|---|---|---|
| N, I/Y, PV, PMT | FV | Input 4 → [CPT] [FV] |
| N, I/Y, PMT, FV | PV | Input 4 → [CPT] [PV] |
| N, I/Y, PV, FV | PMT | PV and FV must have opposite signs |
| PV, PMT, FV | N | Calculate periods; ensure unit consistency |
| N, PV, PMT, FV | I/Y | Result is the periodic rate |
Common Functions
| Operation | Keys |
|---|---|
| Compute eˣ | x [2nd] [LN] |
| Compute ln(x) | x [LN] |
| Compute yˣ | y [yˣ] x [=] |
| Reciprocal | x [1/x] |
| Toggle Sign | [+/-] |
Annuity Quick Reference
| Situation | Mode | Indicator |
|---|---|---|
| Month-end / Year-end | END | "End of month / end of year / arrears" → Ordinary Annuity |
| Month-beginning / Year-beginning | BGN | "Beginning of month / beginning of year / immediate" → Annuity Due |
| Save first, withdraw later | END | Regular savings |
| Pay first, use later | BGN | Rent / lease / lottery winnings |
九、Master Formula Sheet (Full TVM Module)
Single Cash Flow
| Formula | Expression |
|---|---|
| FV (Discrete) | $FV = PV(1 + r)^n$ |
| PV (Discrete) | $PV = FV / (1 + r)^n$ |
| FV (Continuous) | $FV = PV \times e^{r \times n}$ |
| PV (Continuous) | $PV = FV \times e^{-r \times n}$ |
| Simple Interest | $FV = PV(1 + r \times n)$ |
Annuity
| Formula | Expression |
|---|---|
| Ordinary Annuity PV | $PV = PMT \times \frac{1 - (1+r)^{-n}}{r}$ |
| Ordinary Annuity FV | $FV = PMT \times \frac{(1+r)^n - 1}{r}$ |
| Annuity Due PV | $PV = PMT \times \frac{1 - (1+r)^{-n}}{r} \times (1+r)$ |
| Annuity Due FV | $FV = PMT \times \frac{(1+r)^n - 1}{r} \times (1+r)$ |
Perpetuity
| Formula | Expression |
|---|---|
| Ordinary Perpetuity | $PV = PMT / r$ |
| Growing Perpetuity | $PV = PMT_1 / (r - g)$ |
| Delayed Perpetuity | $PV = (PMT / r) / (1+r)^t$ |
Rate Conversions
| Formula | Expression |
|---|---|
| Periodic Rate | $r_{period} = r_s / m$ |
| EAR (m-times compounding) | $EAR = (1 + r_s/m)^m - 1$ |
| EAR (Continuous) | $EAR = e^{r_s} - 1$ |
| Fisher Real Rate | $(1+r_{nom}) = (1+r_{real})(1+\pi)$ |
| Log Return | $r_{log} = \ln(P_t / P_{t-1})$ |
十、Next Lesson Preview
L098 TVM Weekly Quiz — 10 quantitative problems simulating the TVM section of the real CFA exam. Recommended only after achieving ≥ 70% on this practice set.
📊 Core Truth: TVM is the calculation bedrock of CFA Level 1. PV, FV, Annuity, EAR, Continuous Compounding — five keys that open every door. Speed + accuracy = exam-day confidence.