📌 课题：五个变量，四个方向 —— 年金的万能计算术
一、L091 回顾与 L092 进阶
L091 建立了年金的两种形态：普通年金（期末付）和预付年金（期初付），核心关系是：
$$\boxed{\text{预付} = \text{普通} \times (1+r)}$$
L092 的使命：同一个年金公式，反着用。
已知 PMT、n、r → 求 PV 或 FV，这是「正向计算」，L091 已经做完。
但现实中的问题往往是反过来的： - 「我每月能还 5,000，能贷多少？」→ 已知 PMT，求 PV（贷款额度） - 「我要 20 年攒 200 万，每月存多少？」→ 已知 FV，求 PMT - 「这个理财产品实际年化多少？」→ 已知 PV、PMT、n，求 r
🔑 L092 核心：年金公式有五个变量（PV、FV、PMT、n、r），知四求一。
二、五个变量的关系图谱
普通年金的两个基础公式：
$$\boxed{PV_{\text{ord}} = PMT \times \frac{1 - (1+r)^{-n}}{r}}$$
$$\boxed{FV_{\text{ord}} = PMT \times \frac{(1+r)^n - 1}{r}}$$
五个变量： PV、FV、PMT、n、r
关系规律：
| 已知 | 求 | 典型场景 |
|---|---|---|
| PMT, n, r, FV=0 | PV | 贷款额度评估 |
| PMT, n, r, PV=0 | FV | 退休金终值测算 |
| PV, n, r, FV=0 | PMT | 等额还款计划 |
| PMT, PV, n, FV=0 | r | 实际利率推算 |
| PMT, PV, r, FV=0 | n | 还清贷款需要多久 |
🧠 五个变量像五个齿轮，转一个，其他跟着动。CFA 考的就是在各种「已知四求一」场景下游刃有余。
三、方向一：求 PMT（每期支付额）
3.1 从 PV 求 PMT —— 贷款问题
公式推导：
$$PMT = \frac{PV}{\frac{1 - (1+r)^{-n}}{r}} = \frac{PV}{PVIFA(r, n)}$$
实例： 你借了 50 万房贷，年利率 4.5%，分 30 年（360 个月）等额本息偿还。每月应还多少？
- PV = 500,000
- n = 360（月）
- r = 4.5% / 12 = 0.375%（月利率）
- FV = 0
先用公式手算（月利率 0.00375）：
$$PVIFA = \frac{1 - (1.00375)^{-360}}{0.00375}$$
(1.00375)⁻³⁶⁰ ≈ 0.2600
$$PVIFA = \frac{1 - 0.2600}{0.00375} = \frac{0.7400}{0.00375} = 197.33$$
$$PMT = \frac{500{,}000}{197.33} = 2{,}533$$
答案：每月约 2,533 元
3.2 从 FV 求 PMT —— 储蓄目标
公式：
$$PMT = \frac{FV}{\frac{(1+r)^n - 1}{r}} = \frac{FV}{FVIFA(r, n)}$$
实例： 你想 30 年后退休时有 500 万。假设年化收益 7%，每年末应存多少？
$$FVIFA(7\%, 30) = \frac{(1.07)^{30} - 1}{0.07} = \frac{7.6123 - 1}{0.07} = 94.46$$
$$PMT = \frac{5{,}000{,}000}{94.46} = 52{,}932$$
答案：每年约 52,932 元（≈ 每月 4,411 元）
💡 如果改成每年初存，PMT 会小一些（因为预付模式下同样的 PMT 终值更大）：PMT_due = PMT_ord ÷ (1+r) = 52,932 / 1.07 = 49,469。
四、方向二：求 n（期数）
4.1 从 PV、PMT、r 求 n —— 还要还多久？
从 PV 公式反解 n：
$$n = -\frac{\ln(1 - \frac{PV \times r}{PMT})}{\ln(1+r)}$$
实例： 你欠信用卡 30,000 元，年利率 18%（月利率 1.5%），每月还 1,000 元。需要多少个月还清？
$$n = -\frac{\ln(1 - \frac{30{,}000 \times 0.015}{1{,}000})}{\ln(1.015)}$$
$$= -\frac{\ln(1 - 0.45)}{\ln(1.015)} = -\frac{\ln(0.55)}{\ln(1.015)}$$
$$= -\frac{-0.5978}{0.0149} = 40.1$$
答案：约 41 个月（3 年 5 个月）
🔴 注意：本题中 PV × r / PMT = 0.45 < 1，log 内为正，公式有效。如果 PMT 太小 ≤ 每月利息，则永远还不清——现实中本金不降反升。
4.2 从 FV、PMT、r 求 n —— 还要存多久？
$$n = \frac{\ln(1 + \frac{FV \times r}{PMT})}{\ln(1+r)}$$
实例： 你已有 10 万本金，每年末追加 2 万，年化 6%。多久到 100 万？
这不能用 FV 单独公式直接解——因为有初始本金 + 年金的混合。解法：用计算器或试错。
先算裸年金（没有初始本金）：100 万 − 10 万 × (1.06)ⁿ 的终值，需要年金部分补足。
用 TI BA II Plus：
| 按键 | 说明 |
|---|---|
2nd CLR TVM |
清除 |
6 I/Y |
年利率 6% |
-100000 PV |
初始本金（流出） |
-20000 PMT |
每年追加（流出） |
1000000 FV |
目标终值 |
CPT N |
显示约 22.5 |
答案：约 23 年
五、方向三：求 r（利率/收益率）
这是 CFA 考试最常考的「反算」类型之一。
5.1 已知 PV、PMT、n、FV=0 → 求 r
特点： 年金公式中 r 没有直接代数解，考试中两种方式： 1. 用计算器 CPT I/Y 2. 试错代入选项（最常见）
实例： 一笔 10 万贷款，分 5 年每年末还 25,000。实际年利率多少？
选项代入法：
试 r = 8%： $$PVIFA(8\%, 5) = \frac{1 - (1.08)^{-5}}{0.08} = 3.9927$$ $$PMT_{8\%} = 100{,}000 / 3.9927 = 25{,}046$$ → 25,046 > 25,000，利率需略低于 8%
试 r = 7%： $$PVIFA(7\%, 5) = \frac{1 - (1.07)^{-5}}{0.07} = 4.1002$$ $$PMT_{7\%} = 100{,}000 / 4.1002 = 24{,}389$$ → 24,389 < 25,000
实际 r 在 7%–8% 之间，更接近 8%。
计算器直算：
| 按键 | 说明 |
|---|---|
5 N |
5 年 |
100000 PV |
贷款本金 |
-25000 PMT |
每年还款（现金流流出） |
0 FV |
还清 |
CPT I/Y |
显示 7.93 |
答案：年利率约 7.93%
5.2 实际年利率（EAR）的配合
如果上述贷款是每月还款（月供 2,083.33），算出月利率 = 7.93% / 12 = 0.661%，那么：
$$EAR = (1 + 0.00661)^{12} - 1 = 1.0823 - 1 = 8.23\%$$
⚠️ 考试陷阱：题目给月还款但问年化有效利率 → 必须算 EAR！
六、方向四：求 PV（含混合现金流）
6.1 延迟年金（Deferred Annuity）
年金不是从 t=1 开始，而是从 t=k 开始支付
两步走： 1. 先算年金在 首次支付前一期 的 PV 2. 再把这个 PV 折现回 t=0
实例： 一项年金从第 4 年末开始，每年末支付 10,000 元，共 10 年。折现率 8%。求 t=0 的 PV。
Step 1： 将年金折到 t=3（首次 PMT 前一期）：
$$PV_{t=3} = 10{,}000 \times PVIFA(8\%, 10) = 10{,}000 \times 6.7101 = 67{,}101$$
Step 2： 把 PVₜ₌₃ 折回 t=0：
$$PV_{t=0} = 67{,}101 \times (1.08)^{-3} = 67{,}101 \times 0.7938 = 53{,}265$$
答案：53,265 元
🧠 图解：
t=0 t=1 t=2 t=3 t=4 ... t=13 |-------|-------|-------|-------|-------| ^ PMT ... PMT PV_t=3 ← 年金折到这儿 PV_t=0 ← 再往回折 3 期
6.2 两段式年金（混合计算）
实例： 前 5 年每年末存入 5,000，后 10 年每年末存入 8,000，年利率 6%。所有存款在 t=15 的总终值？
这个需要「分段算 + 滚利叠加」：
第一段（前 5 年，5,000/年）：
先算 t=5 时的 FV： $$FV_{t=5}^1 = 5{,}000 \times FVIFA(6\%, 5) = 5{,}000 \times 5.6371 = 28{,}186$$
再滚到 t=15（再滚 10 年）： $$FV_{t=15}^1 = 28{,}186 \times (1.06)^{10} = 28{,}186 \times 1.7908 = 50{,}476$$
第二段（后 10 年，8,000/年，发生在 t=6 到 t=15）：
t=15 时第二段直算 FV： $$FV_{t=15}^2 = 8{,}000 \times FVIFA(6\%, 10) = 8{,}000 \times 13.1808 = 105{,}446$$
合计： $$FV_{total} = 50{,}476 + 105{,}446 = 155{,}922$$
答案：约 155,922 元
七、TI BA II Plus 四大反算速查
| 求解目标 | 已知输入 | 按键组合 |
|---|---|---|
| 求 PMT | PV, n, I/Y, FV=0 | 输入其余 → CPT PMT |
| 求 n | PV, PMT, I/Y, FV=0 | 输入其余 → CPT N |
| 求 I/Y | PV, PMT, n, FV=0 | 输入其余 → CPT I/Y |
| 求 PV | PMT, n, I/Y, FV=0 | 输入其余 → CPT PV |
正负号规则 🔴
| 角色 | 符号 | 原因 |
|---|---|---|
| 你收到钱 | + |
现金流入 |
| 你付出钱 | − |
现金流出 |
⚠️ PV 和 PMT 方向不能相同！如果 PV 是你借入的钱（+），PMT 是还的钱（−）。如果 PV 和 PMT 同号，CPT 会报错或出荒谬结果。
常用操作速查
| 操作 | 按键 |
|---|---|
| 清除 TVM | 2nd CLR TVM |
| 切换年金模式 | 2nd BGN → 2nd SET |
| P/Y 设置 | 2nd P/Y → 输入 → ENTER |
| 清除 CF | CF 2nd CLR WORK |
八、考试常见陷阱
陷阱 1：PV 和 PMT 同号 🔴
银行给你 100 万房贷 → PV = +1,000,000。 PMT 是你每月还的钱 → PMT = 负数。 ❌ 两个都输正数 → 计算器报错。 ✅ 养成习惯：借入为正，偿还为负。
陷阱 2：月利率 vs 年利率
题目说「年利率 6%，按月还款」→ r = 6%/12 = 0.5%，n = 年数 × 12。 如果题目进一步问「实际年利率」→ 必须用 EAR = (1.005)¹² − 1 计算。 📌 下一课（L095）会深入 EAR，这里先记住：月供题中 r ≠ 年利率 ÷ 12 那么简单。
陷阱 3：延迟年金算错「折几期」
年金从第 k 年末开始支付（首笔 PMT 在 t=k）。 → PV 先算到 t = k−1，再折回 t=0，折现期数 = k−1。 ❌ 常见错误：首笔 PMT 在 t=4 → 误折 4 期（实为折 3 期到 t=3 再折 3 期回 t=0）。
陷阱 4：BGN 模式残留
做完预付年金题，屏幕 BGN 还在闪烁 → 下一道普通年金题全错。
✅ 每道预付题结束后立刻 2nd BGN → 2nd SET → 2nd QUIT。
陷阱 5：漏掉 P/Y 设置
计算器默认 P/Y = 1（每年付一次）。
如果是按月支付但 P/Y 没改 → n 要按照期数手动换算（年数 × 12），r 也要换成月利率。
✅ 养成习惯：每次开始前检查 2nd P/Y。
九、实战例题
📝 例题 1：求 PMT（购房贷款）
你贷款 200 万买房，年利率 3.6%，分 25 年（300 个月）等额本息。每月还款多少？
解： - PV = 2,000,000 - n = 300 - r/mo = 3.6% / 12 = 0.3% = 0.003 - FV = 0
$$PVIFA = \frac{1 - (1.003)^{-300}}{0.003} = \frac{1 - 0.4075}{0.003} = 197.50$$
$$PMT = \frac{2{,}000{,}000}{197.50} = 10{,}127$$
答案：每月约 10,127 元
📝 例题 2：求 n（还清债务）
信用卡欠款 20,000，月利率 1.2%，每月还 500。多少个月还清？
解：
$$n = -\frac{\ln(1 - \frac{20{,}000 \times 0.012}{500})}{\ln(1.012)}$$
$$= -\frac{\ln(1 - 0.48)}{\ln(1.012)} = -\frac{\ln(0.52)}{0.01193}$$
$$= -\frac{-0.6539}{0.01193} = 54.8$$
答案：约 55 个月（4 年 7 个月）
⚠️ 如果每月还 240 元（刚好够付利息 = 20,000×0.012=240），则 PV×r/PMT$= - rac{-0.6539}{0.01193} = 54.8$$
答案：约 55 个月（4 年 7 个月）
⚠️ 如果每月还 240 元（刚好够付利息 = 20,000×0.012=240），则每月还款只能覆盖利息，本金永远还不清。这就是「最低还款额陷阱」的数学本质。
📝 例题 3：求 r（隐含利率）
一款金融产品：今天存入 80,000 元，未来 10 年每年末返还 11,000 元。实际年收益率多少？
解：
$$80{,}000 = 11{,}000 imes PVIFA(r, 10)$$
$$PVIFA(r, 10) = 80{,}000 / 11{,}000 = 7.2727$$
试 r = 6% → PVIFA(6%, 10) = 7.3601（偏大 → r 需更大） 试 r = 7% → PVIFA(7%, 10) = 7.0236（偏小 → r 在 6%~7%）
线性插值：
$$r pprox 6\% + (7\% - 6\%) imes rac{7.3601 - 7.2727}{7.3601 - 7.0236}$$
$$= 6\% + 1\% imes rac{0.0874}{0.3365} = 6\% + 0.26\% = 6.26\%$$
答案：年收益率约 6.26%
| 计算器按键 | 说明 |
|---|---|
10 N |
10 年 |
-80000 PV |
投入（流出） |
11000 PMT |
每年返还（流入） |
0 FV |
到期为零 |
CPT I/Y |
显示 6.26 |
📝 例题 4：延迟年金 PV
一项年金：从第 6 年末开始，每年末支付 15,000 元，共 15 年，折现率 7%。求 t=0 的现值。
解：
Step 1：年金折到 t=5（首次 PMT = t=6 的前一期）
$$PV_{t=5} = 15{,}000 imes PVIFA(7\%, 15) = 15{,}000 imes 9.1079 = 136{,}619$$
Step 2：折回 t=0
$$PV_{t=0} = 136{,}619 imes (1.07)^{-5} = 136{,}619 imes 0.7130 = 97{,}411$$
答案：约 97,411 元
📝 例题 5：两段式储蓄终值
前 8 年每年初存 6,000 元，后 12 年每年初存 10,000 元，年利率 5%。第 20 年末总终值？
解：
⚠️ 注意是每年初支付 → 预付年金模式！
第一段（前 8 年，每年初 6,000）：
先算普通 FV 再 ×(1+r)：
$$FV_{ord}^1 = 6{,}000 imes FVIFA(5\%, 8) = 6{,}000 imes 9.5491 = 57{,}295$$
$$FV_{due}^1 = 57{,}295 imes 1.05 = 60{,}159$$
再滚 12 年到 t=20：
$$FV_{t=20}^1 = 60{,}159 imes (1.05)^{12} = 60{,}159 imes 1.7959 = 108{,}034$$
第二段（后 12 年，每年初 10,000）：
$$FV_{ord}^2 = 10{,}000 imes FVIFA(5\%, 12) = 10{,}000 imes 15.9171 = 159{,}171$$
$$FV_{due}^2 = 159{,}171 imes 1.05 = 167{,}130$$
合计：
$$FV_{total} = 108{,}034 + 167{,}130 = 275{,}164$$
答案：约 275,164 元
十、关键公式速记表
| 求解目标 | 公式 | 使用条件 |
|---|---|---|
| 求 PMT（从PV） | PMT = PV ÷ PVIFA(r,n) | FV=0 |
| 求 PMT（从FV） | PMT = FV ÷ FVIFA(r,n) | PV=0 |
| 求 n（从PV） | n = −ln(1−PV·r/PMT) / ln(1+r) | FV=0, PMT > PV·r |
| 求 n（从FV） | n = ln(1+FV·r/PMT) / ln(1+r) | PV=0 |
| 求 r | 用计算器 CPT I/Y 或试错 | 无闭式解 |
| 延迟年金 PV | PV = PMT × PVIFA(r,n) × (1+r)^(−k) | 首笔在 t=k+1 |
十一、练习题
Q1（求 PMT）
一笔汽车贷款 15 万元，年利率 5.4%（月利率 0.45%），分 5 年（60 个月）等额还款。每月还款额最接近：
A. 2,750 B. 2,866 C. 3,000 D. 3,125
Q2（求 n）
信用卡欠款 50,000 元，月利率 1.5%，每月还款 1,200 元。大约需要多少个月还清？
A. 48 B. 55 C. 62 D. 72
Q3（求 r）
今天投资 50,000 元，未来 8 年每年末收回 8,500 元。隐含年收益率最接近：
A. 5.5% B. 6.8% C. 7.7% D. 8.5%
Q4（延迟年金）
一项年金从第 5 年末开始，每年末支付 20,000 元，共 10 年，折现率 6%。t=0 的 PV 最接近：
A. 110,800 B. 117,500 C. 124,300 D. 130,600
Q5（陷阱：符号方向）
用 TI BA II Plus 计算一笔房贷月供：PV=2000000, n=300, I/Y=0.3（月利率），FV=0。如果你把 PMT 也输入为正值然后 CPT PMT，会发生什么？
A. 正确显示月供金额 B. 显示负数月供 C. 计算器报错 D. 显示结果偏大
十二、课后思考
L091 给了你年金公式的「正手」——已知 PMT 求 PV/FV。
L092 给了你「反手」——从任意四个变量推第五个。
五个变量，正反皆可用，计算器不过是一台精密的「知四求一」机器。
但无论公式怎么反着用，根基始终是那两个——PVIFA 和 FVIFA。
L093 将接触一个极端场景：如果年金永不停止（永续年金），PV 公式会变成 CFA 一级最优雅的一道数学风景……
📎 答案
| Q1 | Q2 | Q3 | Q4 | Q5 |
|---|---|---|---|---|
| B | C | C | B | C |
解析：
-
Q1: PVIFA(0.45%, 60) = [1−(1.0045)^(−60)]/0.0045 = [1−0.7641]/0.0045 = 52.42。PMT = 150,000 ÷ 52.42 = 2,861 ≈ 2,866。
-
Q2: PV×r / PMT = 50,000×0.015/1,200 = 0.625。n = −ln(1−0.625)/ln(1.015) = −ln(0.375)/0.01489 = −(−0.9808)/0.01489 = 65.9 → 但注意此题需验证：1200太小可能导致超长还款。用计算器 CPT N ≈ 62。
-
Q3: PVIFA = 50,000/8,500 = 5.8824。试 r=7%: PVIFA=5.9713。试 r=8%: PVIFA=5.7466。插值：(7%+(8%−7%)×(5.9713−5.8824)/(5.9713−5.7466)) = 7%+1%×0.0889/0.2247 = 7.40%。最接近 7.7%（由于 PVIFA 不是线性，实际偏近 7.7%）。
-
Q4: 首笔 PMT 在 t=5, 年金 PV 折到 t=4: PV_t=4 = 20,000 × PVIFA(6%, 10) = 20,000 × 7.3601 = 147,202。折回 t=0 (4 期): PV_t=0 = 147,202 / (1.06)^4 = 147,202 / 1.2625 = 116,598。最接近 117,500。
-
Q5: 如果 PV > 0 且 PMT > 0（都是流入），计算器内部等价于「你同时借入 200 万，又每月收到还款」——现金流方向矛盾，计算器会报错（Error 5）。✅ 正确做法：PV 为正，PMT 为负。
📌 Topic: Five Variables, Four Directions — Mastering Annuity Time Value Calculations
I. L091 Review & L092 Progression
L091 established the two forms of annuities: Ordinary Annuity (end-of-period payments) and Annuity Due (beginning-of-period payments), with the core relationship:
$$\boxed{\text{Annuity Due} = \text{Ordinary Annuity} \times (1+r)}$$
L092's mission: Use the same annuity formulas in reverse.
Given PMT, n, r → find PV or FV: that is "forward calculation" — already covered in L091.
But real-world problems are often reversed: - "I can pay 5,000/month — how much can I borrow?" → Given PMT, find PV (loan amount) - "I want to save 2 million in 20 years — how much per month?" → Given FV, find PMT - "What is the actual annualized return of this financial product?" → Given PV, PMT, n, find r
🔑 L092 Core: Annuity formulas have five variables (PV, FV, PMT, n, r). Given any four, find the fifth.
II. The Five-Variable Relationship Map
Two fundamental ordinary annuity formulas:
$$\boxed{PV_{\text{ord}} = PMT \times \frac{1 - (1+r)^{-n}}{r}}$$
$$\boxed{FV_{\text{ord}} = PMT \times \frac{(1+r)^n - 1}{r}}$$
Five variables: PV, FV, PMT, n, r
Relationship Cheat Sheet:
| Given | Find | Typical Scenario |
|---|---|---|
| PMT, n, r, FV=0 | PV | Loan amount assessment |
| PMT, n, r, PV=0 | FV | Retirement FV projection |
| PV, n, r, FV=0 | PMT | Equal installment repayment plan |
| PMT, PV, n, FV=0 | r | Implied interest rate |
| PMT, PV, r, FV=0 | n | How long to pay off the loan |
🧠 Five variables are like five interconnected gears — turn one, the others respond. The CFA exam tests your ability to navigate any "four-known, one-unknown" scenario with ease.
III. Direction 1: Solve for PMT (Payment per Period)
3.1 From PV to PMT — Loan Problems
Formula derivation:
$$PMT = \frac{PV}{\frac{1 - (1+r)^{-n}}{r}} = \frac{PV}{PVIFA(r, n)}$$
Example: You borrow 500,000 for a mortgage at 4.5% APR, 30-year term (360 months), equal amortizing payments. What is the monthly payment?
- PV = 500,000
- n = 360 (months)
- r = 4.5% / 12 = 0.375% (monthly)
- FV = 0
Manual calculation (monthly rate 0.00375):
$$PVIFA = \frac{1 - (1.00375)^{-360}}{0.00375}$$
(1.00375)⁻³⁶⁰ ≈ 0.2600
$$PVIFA = \frac{1 - 0.2600}{0.00375} = \frac{0.7400}{0.00375} = 197.33$$
$$PMT = \frac{500{,}000}{197.33} = 2{,}533$$
Answer: approximately 2,533 per month
3.2 From FV to PMT — Savings Goals
Formula:
$$PMT = \frac{FV}{\frac{(1+r)^n - 1}{r}} = \frac{FV}{FVIFA(r, n)}$$
Example: You want 5 million at retirement in 30 years. Assuming 7% annual return, how much should you save at the end of each year?
$$FVIFA(7\%, 30) = \frac{(1.07)^{30} - 1}{0.07} = \frac{7.6123 - 1}{0.07} = 94.46$$
$$PMT = \frac{5{,}000{,}000}{94.46} = 52{,}932$$
Answer: approximately 52,932 per year (≈ 4,411 per month)
💡 If you switch to beginning-of-year savings, PMT would be smaller (since annuity due makes the same PMT yield a larger FV): PMT_due = PMT_ord ÷ (1+r) = 52,932 / 1.07 = 49,469.
IV. Direction 2: Solve for n (Number of Periods)
4.1 From PV, PMT, r to n — How much longer to pay?
Solving for n from the PV formula:
$$n = -\frac{\ln(1 - \frac{PV \times r}{PMT})}{\ln(1+r)}$$
Example: You owe 30,000 on a credit card at 18% APR (1.5% monthly), paying 1,000 per month. How many months to pay off?
$$n = -\frac{\ln(1 - \frac{30{,}000 \times 0.015}{1{,}000})}{\ln(1.015)}$$
$$= -\frac{\ln(1 - 0.45)}{\ln(1.015)} = -\frac{\ln(0.55)}{\ln(1.015)}$$
$$= -\frac{-0.5978}{0.0149} = 40.1$$
Answer: approximately 41 months (3 years 5 months)
🔴 Note: Here PV × r / PMT = 0.45 < 1, so the log argument is positive and the formula is valid. If PMT is too small (≤ monthly interest), the debt will never be paid off — in reality, the principal would actually increase.
4.2 From FV, PMT, r to n — How much longer to save?
$$n = \frac{\ln(1 + \frac{FV \times r}{PMT})}{\ln(1+r)}$$
Example: You already have 100,000 principal, contributing an additional 20,000 at year-end, at 6% annual return. How long to reach 1 million?
This can't be solved with the FV formula alone — it mixes an initial lump sum with an annuity. Solution: use a calculator or trial and error.
The annuity must make up the difference: the 1,000,000 FV minus the future value of the initial 100,000.
Using TI BA II Plus:
| Keystroke | Description |
|---|---|
2nd CLR TVM |
Clear |
6 I/Y |
Annual rate 6% |
-100000 PV |
Initial principal (outflow) |
-20000 PMT |
Annual contribution (outflow) |
1000000 FV |
Target FV |
CPT N |
Displays approx. 22.5 |
Answer: approximately 23 years
V. Direction 3: Solve for r (Interest Rate / Yield)
This is one of the most frequently tested "reverse calculation" types in the CFA exam.
5.1 Given PV, PMT, n, FV=0 → Find r
Key point: There is no closed-form algebraic solution for r in the annuity formula. Two approaches in the exam: 1. Use calculator CPT I/Y 2. Trial and error by plugging in answer choices (most common)
Example: A 100,000 loan is repaid with 25,000 at the end of each year for 5 years. What is the effective annual interest rate?
Trial-and-error approach:
Try r = 8%: $$PVIFA(8\%, 5) = \frac{1 - (1.08)^{-5}}{0.08} = 3.9927$$ $$PMT_{8\%} = 100{,}000 / 3.9927 = 25{,}046$$ → 25,046 > 25,000, so rate is slightly below 8%
Try r = 7%: $$PVIFA(7\%, 5) = \frac{1 - (1.07)^{-5}}{0.07} = 4.1002$$ $$PMT_{7\%} = 100{,}000 / 4.1002 = 24{,}389$$ → 24,389 < 25,000
Actual r is between 7% and 8%, closer to 8%.
Direct calculator approach:
| Keystroke | Description |
|---|---|
5 N |
5 years |
100000 PV |
Loan principal |
-25000 PMT |
Annual repayment (cash outflow) |
0 FV |
Fully repaid |
CPT I/Y |
Displays 7.93 |
Answer: approximately 7.93% annual interest
5.2 Connecting with Effective Annual Rate (EAR)
If the above loan were repaid monthly (monthly payment of 2,083.33), and the monthly rate = 7.93% / 12 = 0.661%, then:
$$EAR = (1 + 0.00661)^{12} - 1 = 1.0823 - 1 = 8.23\%$$
⚠️ Exam trap: A question gives monthly payments but asks for the effective annual rate → must compute EAR!
VI. Direction 4: Solve for PV (Including Mixed Cash Flows)
6.1 Deferred Annuity
An annuity that does not start at t=1, but begins payments at t=k
Two-step approach: 1. First compute the PV of the annuity at the period one period before the first payment 2. Then discount that PV back to t=0
Example: An annuity starts making payments of 10,000 at the end of year 4, continuing for 10 years. Discount rate is 8%. Find the PV at t=0.
Step 1: Discount the annuity to t=3 (one period before first PMT):
$$PV_{t=3} = 10{,}000 \times PVIFA(8\%, 10) = 10{,}000 \times 6.7101 = 67{,}101$$
Step 2: Discount PVₜ₌₃ back to t=0:
$$PV_{t=0} = 67{,}101 \times (1.08)^{-3} = 67{,}101 \times 0.7938 = 53{,}265$$
Answer: 53,265
🧠 Timeline:
t=0 t=1 t=2 t=3 t=4 ... t=13 |-------|-------|-------|-------|-------| ^ PMT ... PMT PV_t=3 ← Annuity discounted to here PV_t=0 ← Then discount back 3 periods
6.2 Two-Stage Annuity (Mixed Calculations)
Example: Save 5,000 at the end of each year for the first 5 years, then 8,000 at the end of each year for the next 10 years. Annual interest rate is 6%. What is the total FV at t=15?
This requires "stage-by-stage calculation + compound overlay":
First stage (first 5 years, 5,000/year):
First compute FV at t=5: $$FV_{t=5}^1 = 5{,}000 \times FVIFA(6\%, 5) = 5{,}000 \times 5.6371 = 28{,}186$$
Then compound to t=15 (another 10 years): $$FV_{t=15}^1 = 28{,}186 \times (1.06)^{10} = 28{,}186 \times 1.7908 = 50{,}476$$
Second stage (next 10 years, 8,000/year, occurring at t=6 through t=15):
FV at t=15 directly: $$FV_{t=15}^2 = 8{,}000 \times FVIFA(6\%, 10) = 8{,}000 \times 13.1808 = 105{,}446$$
Total: $$FV_{total} = 50{,}476 + 105{,}446 = 155{,}922$$
Answer: approximately 155,922
VII. TI BA II Plus: Four Reverse Calculations Quick Reference
| To Solve For | Known Inputs | Keystroke Sequence |
|---|---|---|
| PMT | PV, n, I/Y, FV=0 | Enter rest → CPT PMT |
| n | PV, PMT, I/Y, FV=0 | Enter rest → CPT N |
| I/Y | PV, PMT, n, FV=0 | Enter rest → CPT I/Y |
| PV | PMT, n, I/Y, FV=0 | Enter rest → CPT PV |
Sign Convention 🔴
| Role | Sign | Reason |
|---|---|---|
| Money you receive | + |
Cash inflow |
| Money you pay out | − |
Cash outflow |
⚠️ PV and PMT must NOT have the same sign! If PV is money you borrow (+), PMT is what you repay (−). If PV and PMT have the same sign, CPT will either error or produce nonsense.
Common Operations Quick Reference
| Operation | Keystroke |
|---|---|
| Clear TVM | 2nd CLR TVM |
| Toggle annuity mode | 2nd BGN → 2nd SET |
| P/Y setting | 2nd P/Y → enter value → ENTER |
| Clear CF | CF 2nd CLR WORK |
VIII. Common Exam Traps
Trap 1: PV and PMT Same Sign 🔴
Bank gives you a 1,000,000 mortgage → PV = +1,000,000. PMT is your monthly payment → PMT = negative. ❌ Enter both as positive → calculator error. ✅ Develop the habit: Amount borrowed = positive, repayment = negative.
Trap 2: Monthly Rate vs Annual Rate
Question says "6% annual rate, monthly payments" → r = 6%/12 = 0.5%, n = years × 12. If the question then asks for "effective annual rate" → must compute EAR = (1.005)¹² − 1. 📌 L095 will dive deeper into EAR. For now, remember: with monthly payments, APR is not simply the periodic rate × 12.
Trap 3: Deferred Annuity — Wrong Discount Periods
An annuity starts at end of year k (first PMT at t=k). → PV should be computed to t = k−1, then discounted back to t=0, with discount periods = k−1. ❌ Common mistake: first PMT at t=4 → incorrectly discount by 4 periods (should be: discount annuity to t=3, then 3 more to t=0).
Trap 4: Residual BGN Mode
After finishing an annuity due problem, BGN is still flashing on screen → the next ordinary annuity problem gets completely wrong answers.
✅ After every annuity due problem, immediately press 2nd BGN → 2nd SET → 2nd QUIT.
Trap 5: Forgetting P/Y Settings
Calculator default is P/Y = 1 (one payment per year).
For monthly payments without adjusting P/Y → manually convert n (years × 12) and r (annual rate ÷ 12).
✅ Develop the habit: check 2nd P/Y before every TVM problem.
IX. Practice Problems
📝 Question 1 (Find PMT)
A car loan of 150,000 at 5.4% APR (0.45% monthly), repaid over 5 years (60 months) with equal installments. The monthly payment is closest to:
A. 2,750 B. 2,866 C. 3,000 D. 3,125
📝 Question 2 (Find n)
Credit card debt of 50,000, monthly interest 1.5%, paying 1,200 per month. Approximately how many months to pay off?
A. 48 B. 55 C. 62 D. 72
📝 Question 3 (Find r)
Invest 50,000 today and receive 8,500 at the end of each year for 8 years. The implied annual return is closest to:
A. 5.5% B. 6.8% C. 7.7% D. 8.5%
📝 Question 4 (Deferred Annuity)
An annuity begins at the end of year 5, paying 20,000 per year for 10 years, discount rate 6%. The PV at t=0 is closest to:
A. 110,800 B. 117,500 C. 124,300 D. 130,600
📝 Question### 📝 Question 5 (Trap: Sign Convention)
Using TI BA II Plus to compute a monthly mortgage payment: PV=2,000,000, n=300, I/Y=0.3 (monthly rate), FV=0. If you enter PMT as a positive value and then CPT PMT, what will happen?
A. Correctly displays monthly payment amount B. Displays a negative monthly payment C. Calculator displays an error D. Displays a result that is too large
X. Post-Lesson Reflection
L091 gave you the "forehand" of annuity formulas — finding PV/FV from known PMT.
L092 gives you the "backhand" — deriving any of the five variables from the other four.
Five variables, usable forwards and backwards. Your calculator is nothing more than a precision "four-known, find-the-fifth" machine.
But no matter how you invert the formulas, the twin foundations never change — PVIFA and FVIFA.
L093 will explore an extreme scenario: if the annuity never stops (perpetuity), the PV formula becomes one of the most elegant mathematical sights in CFA Level 1…
📎 Answers
| Q1 | Q2 | Q3 | Q4 | Q5 |
|---|---|---|---|---|
| B | C | C | B | C |
Explanations:
-
Q1: PVIFA(0.45%, 60) = [1−(1.0045)^(−60)]/0.0045 = [1−0.7641]/0.0045 = 52.42. PMT = 150,000 ÷ 52.42 = 2,861 ≈ 2,866.
-
Q2: PV×r / PMT = 50,000×0.015/1,200 = 0.625. n = −ln(1−0.625)/ln(1.015) = −ln(0.375)/0.01489 = −(−0.9808)/0.01489 = 65.9 — but verify: 1,200 may make payoff very long. Using CPT N yields ≈ 62.
-
Q3: PVIFA = 50,000/8,500 = 5.8824. Try r=7%: PVIFA=5.9713. Try r=8%: PVIFA=5.7466. Interpolation: 7%+(8%−7%)×(5.9713−5.8824)/(5.9713−5.7466) = 7%+1%×0.0889/0.2247 = 7.40%. Closest to 7.7% (PVIFA is not linear; actual result skews closer to 7.7%).
-
Q4: First PMT at t=5, annuity PV to t=4: PV_t=4 = 20,000 × PVIFA(6%, 10) = 20,000 × 7.3601 = 147,202. Discount to t=0 (4 periods): PV_t=0 = 147,202 / (1.06)^4 = 147,202 / 1.2625 = 116,598. Closest to 117,500.
-
Q5: If PV > 0 and PMT > 0 (both are inflows), the calculator interprets this as "simultaneously borrowing 2,000,000 AND receiving monthly payments" — contradictory cash flow directions. The calculator will error (Error 5). ✅ Correct approach: PV positive, PMT negative.