📌 课题：现值 PV 计算 —— 把未来的钱"折"回今天
一、回顾与引入
L087 我们认识了 PV（现值）与 FV（终值）这对 TVM 的核心概念。L088 又深入比较了单利与复利的本质区别——复利让利息生息，单利则只在本金上线性增长。
本课把这两条线合二为一：给定未来的单笔现金流，如何精确计算它的现值 PV？
🔑 核心思维转换：现值计算 = "往回折现"——把终值 FV 乘以一个小于 1 的折现因子，得到今天的等值金额。
二、核心公式推导
2.1 从 FV 公式反推 PV
L087 中我们学过复利终值公式：
$$FV = PV \times (1 + r)^n$$
移项，直接得到 PV 公式：
$$\boxed{PV = \frac{FV}{(1 + r)^n}}$$
另一种书写方式（等价）：
$$PV = FV \times (1 + r)^{-n}$$
参数说明：
- FV = 终值（未来某时刻的现金流金额）
- r = 每期利率（折现率 / discount rate）
- n = 期数
- (1 + r)^{-n} = 折现因子（Discount Factor）
🧠 直觉记忆：PV < FV（正利率下）——今天的 1 元比明天的 1 元值钱。折现率 r 越高、期数 n 越长，PV 就越小。
2.2 单利下的 PV 计算
若现金流按单利计息，PV 公式为：
$$PV = \frac{FV}{1 + r \times n}$$
⚠️ 一级考试中，除非题目明确指定「单利」，否则默认使用复利折现。
三、实战例题
📝 例题 1：基础复利折现
你 5 年后需要 100,000 元，市场年利率为 6%，按年复利。现在需要存入多少？
解：
$$PV = \frac{100{,}000}{(1 + 0.06)^5} = \frac{100{,}000}{1.3382} = 74{,}725.82$$
验算：74,725.82 × (1.06)⁵ = 74,725.82 × 1.3382 ≈ 100,000 ✓
答案：74,726 元（约）
📝 例题 2：不同折现率的比较
同一笔未来现金流 FV = 10,000，n = 3 年。比较 r = 5% vs r = 10% 下的 PV。
| 折现率 r | 折现因子 (1+r)⁻³ | PV | 变化 |
|---|---|---|---|
| 5% | 1.05⁻³ = 0.8638 | 8,638 | — |
| 10% | 1.10⁻³ = 0.7513 | 7,513 | ↓ 1,125 |
🔑 折现率越高 → PV 越低。这反映了「资金的机会成本」——如果你的钱能在别处赚到更高回报，未来同样金额对你今天的价值就更低。
📝 例题 3：非整数年期（季度复利）
你 2.5 年后收到 50,000 元，年利率 8%，按半年复利（即每半年计息一次）。求 PV。
关键：调整 r 和 n 以匹配复利频率！
- 每期利率：
r_period = 8% ÷ 2 = 4%（半年利率） - 期数：
n = 2.5 × 2 = 5（个半年） - PV = 50,000 ÷ (1.04)⁵ = 50,000 ÷ 1.2167 = 41,095
答案：41,095 元
⚠️ 这是「调频」问题的起点——L095 会系统讲解名义利率 vs 有效年利率（EAR），这里先建立直觉：复利频率越高，同样名义利率下折现越深，PV 越小。
📝 例题 4：用 PV 判断投资机会
某投资项目承诺：3 年后一次性返还 130,000 元，现在你需要投入 100,000 元。你的要求回报率为 8%。该投吗？
分析思路：把未来回报折现，与当前投入比较。
$$PV_{回报} = \frac{130{,}000}{(1.08)^3} = \frac{130{,}000}{1.2597} = 103{,}212$$
PV（103,212）> 投入（100,000）→ 净现值 NPV = 3,212 > 0 → 值得投资。
四、折现因子速查表（常用 r × n 组合）
| r \ n | 1 年 | 3 年 | 5 年 | 10 年 |
|---|---|---|---|---|
| 5% | 0.9524 | 0.8638 | 0.7835 | 0.6139 |
| 8% | 0.9259 | 0.7938 | 0.6806 | 0.4632 |
| 10% | 0.9091 | 0.7513 | 0.6209 | 0.3855 |
🧠 记住两个经典数字：10%/5 年的折现因子 ≈ 0.62，10%/10 年 ≈ 0.39——未来越远，价值衰减越快。
五、常见计算器操作（BA II Plus / HP 12C）
| 步骤 | BA II Plus | 说明 |
|---|---|---|
| 清零 | [2nd] [FV] (CLR TVM) |
消除之前 TVM 变量 |
| 输入 FV | 100000 [FV] |
未来现金流 |
| 输入 r | 6 [I/Y] |
年利率（%） |
| 输入 n | 5 [N] |
期数 |
| 设 PMT=0 | 0 [PMT] |
无期间现金流 |
| 计算 PV | [CPT] [PV] |
得出 -74,725.82 |
BA II Plus 默认 PV 和 FV 符号相反（一进一出），负号表示现金流出。
六、常见错误与陷阱 ⚠️
陷阱 1：忘记调整复利频率
题目说「半年复利」，你却用年化 r 和年数 n 直接算——必错。 口诀：r 和 n 必须匹配同一时间单位！
陷阱 2：单利/复利混淆
「存款 3 年，年利率 5%」没说明单利/复利时，默认复利。但债券的应计利息、某些短期票据可能用单利——盯紧题目表述。
陷阱 3：符号方向
BA II Plus 中 PV 和 FV 必然正负相反。如果 PV 和 FV 同号 → 你的现金流方向设错了。
陷阱 4：PV 不是「原值」
PV ≠ 投入本金。PV 是「未来现金流在今天的等值金额」，是一个计算结果。如果你先投入 P₀，再计算未来回报的 PV，二者比较才有意义（如例题 4 的 NPV 思路）。
七、核心公式速记卡
| 条件 | 公式 | 备注 |
|---|---|---|
| 复利 PV（常规） | PV = FV / (1+r)ⁿ | 🔥 最常用 |
| 复利 PV（等价写法） | PV = FV × (1+r)⁻ⁿ | 折现因子形式 |
| 单利 PV | PV = FV / (1+r×n) | 仅题目明确单利时用 |
| 非年复利 | r_per = r/m, n_per = n×m | m = 每年复利次数 |
📝 测试题
Q1（基础概念）
今天存入一笔钱，年利率 8%，按年复利，5 年后获得 50,000 元。现在需要存入多少？（四舍五入取整）
A. 30,000 元 B. 34,029 元 C. 36,050 元 D. 40,000 元
Q2（折现率比较）
未来 10 年后的 10,000 元在以下哪个折现率下 PV 最高？
A. r = 5% B. r = 8% C. r = 10% D. r = 12%
Q3（复利频率陷阱）
一笔未来现金流 FV = 20,000，n = 2 年，年利率 10%，按季度复利。PV 最接近：
A. 16,200 B. 16,400 C. 16,529 D. 17,000
Q4（NPV 判断）
某投资项目需现在投入 50,000 元，3 年后一次性回报 65,000 元。你的要求回报率为 10%。下列判断正确的是：
A. NPV > 0，应该投资 B. NPV < 0，不应该投资 C. NPV = 0，投不投无差异 D. 信息不足，无法判断
Q5（单利 vs 复利）
同一笔 FV = 10,000，n = 4 年，r = 5%。单利折现下的 PV 比复利折现下的 PV：
A. 更大 B. 更小 C. 相等 D. 无法确定
📋 答案
| 题号 | 答案 | 解析 |
|---|---|---|
| Q1 | B | PV = 50,000 / (1.08)⁵ = 50,000 / 1.4693 = 34,029 |
| Q2 | A | 折现率越低，PV 越大。r=5% 时折现因子最大 |
| Q3 | C | r_per = 10%/4 = 2.5%, n = 8 → PV = 20,000 / (1.025)⁸ = 20,000 / 1.2184 = 16,529 |
| Q4 | B | PV = 65,000 / 1.1³ = 65,000 / 1.331 = 48,835 < 50,000 → NPV = -1,165 < 0 |
| Q5 | A | 单利折现因子 = 1/(1+0.05×4) = 1/1.20 = 0.8333；复利 = 1/(1.05)⁴ = 0.8227 → 单利 PV 更大 |
🎯 掌握 PV 计算，你就拿到了 TVM 世界的「折现」钥匙。下节课 L090 将学习 FV 计算（单笔现金流），两个方向都熟练了，才能自如地穿越时间。
📌 Topic: Present Value Calculation — Discounting Future Cash Flows Back to Today
1. Review & Introduction
In L087, we learned about TVM's two core concepts: Present Value (PV) and Future Value (FV), along with the interest rate that bridges them. In L088, we compared Simple vs. Compound Interest — compound interest generates "interest on interest," while simple interest grows linearly on principal only.
This lesson merges both threads: given a single future cash flow, how do we precisely calculate its Present Value?
🔑 Key mindset shift: PV calculation = "discounting backwards" — multiplying the FV by a discount factor (< 1) to get today's equivalent amount.
2. Core Formula Derivation
2.1 Deriving PV from the FV Formula
From L087, the compound interest FV formula is:
$$FV = PV \times (1 + r)^n$$
Rearranging gives the PV formula:
$$\boxed{PV = \frac{FV}{(1 + r)^n}}$$
Equivalent form:
$$PV = FV \times (1 + r)^{-n}$$
Parameters:
- FV = Future Value (the cash flow amount at a future point in time)
- r = Periodic interest rate (discount rate)
- n = Number of periods
- (1 + r)^{-n} = Discount Factor
🧠 Intuition: PV < FV (under positive rates) — $1 today is worth more than $1 tomorrow. The higher the discount rate r and the longer the horizon n, the smaller the PV.
2.2 PV Under Simple Interest
If cash flows use simple interest, the PV formula is:
$$PV = \frac{FV}{1 + r \times n}$$
⚠️ For Level 1, unless the problem explicitly states "simple interest," always default to compound discounting.
3. Worked Examples
📝 Example 1: Basic Compound Discounting
You need $100,000 in 5 years. The market annual rate is 6%, compounded annually. How much do you need to deposit today?
Solution:
$$PV = \frac{100{,}000}{(1 + 0.06)^5} = \frac{100{,}000}{1.3382} = 74{,}725.82$$
Check: 74,725.82 × (1.06)⁵ = 74,725.82 × 1.3382 ≈ 100,000 ✓
Answer: $74,726 (approx.)
📝 Example 2: Comparing Different Discount Rates
Same future cash flow FV = $10,000, n = 3 years. Compare PV under r = 5% vs r = 10%.
| Discount Rate r | Discount Factor (1+r)⁻³ | PV | Change |
|---|---|---|---|
| 5% | 1.05⁻³ = 0.8638 | $8,638 | — |
| 10% | 1.10⁻³ = 0.7513 | $7,513 | ↓ $1,125 |
🔑 Higher discount rate → Lower PV. This reflects the opportunity cost of capital — if your money can earn higher returns elsewhere, the same future amount is worth less to you today.
📝 Example 3: Non-Integer Periods (Semi-Annual Compounding)
You will receive $50,000 in 2.5 years. Annual rate is 8%, compounded semi-annually. Find PV.
Key: Adjust r and n to match the compounding frequency!
- Per-period rate:
r_period = 8% ÷ 2 = 4%(semi-annual rate) - Number of periods:
n = 2.5 × 2 = 5half-years - PV = 50,000 ÷ (1.04)⁵ = 50,000 ÷ 1.2167 = $41,095
Answer: $41,095
⚠️ This is the starting point for "frequency adjustment" problems — L095 will systematically cover nominal vs. effective annual rate (EAR). For now, build the intuition: the higher the compounding frequency, the deeper the discounting at the same nominal rate, and the smaller the PV.
📝 Example 4: Using PV to Evaluate an Investment
An investment promises to pay $130,000 in 3 years. You need to invest $100,000 now. Your required rate of return is 8%. Should you invest?
Analysis: Discount the future payoff and compare with the current outlay.
$$PV_{payoff} = \frac{130{,}000}{(1.08)^3} = \frac{130{,}000}{1.2597} = 103{,}212$$
PV ($103,212) > Investment ($100,000) → NPV = $3,212 > 0 → Worth investing.
4. Discount Factor Quick Reference (Common r × n Combinations)
| r \ n | 1 yr | 3 yr | 5 yr | 10 yr |
|---|---|---|---|---|
| 5% | 0.9524 | 0.8638 | 0.7835 | 0.6139 |
| 8% | 0.9259 | 0.7938 | 0.6806 | 0.4632 |
| 10% | 0.9091 | 0.7513 | 0.6209 | 0.3855 |
🧠 Remember two classic numbers: 10%/5-year discount factor ≈ 0.62; 10%/10-year ≈ 0.39 — the further into the future, the faster value decays.
5. Common Calculator Operations (BA II Plus / HP 12C)
| Step | BA II Plus | Explanation |
|---|---|---|
| Clear | [2nd] [FV] (CLR TVM) |
Clear previous TVM variables |
| Enter FV | 100000 [FV] |
Future cash flow |
| Enter r | 6 [I/Y] |
Annual rate (%) |
| Enter n | 5 [N] |
Number of periods |
| Set PMT=0 | 0 [PMT] |
No intermediate cash flows |
| Compute PV | [CPT] [PV] |
Returns -74,725.82 |
BA II Plus defaults to PV and FV having opposite signs (inflow vs. outflow). The negative sign indicates a cash outflow.
6. Common Mistakes & Pitfalls ⚠️
Pitfall 1: Forgetting to Adjust Compounding Frequency
Problem says "semi-annual compounding," but you use the annual rate r directly with annual n — guaranteed to be wrong. Rule of thumb: r and n must match the same time unit!
Pitfall 2: Confusing Simple vs. Compound Interest
"Deposit for 3 years at 5% annual rate" without specifying simple/compound → default to compound. However, bond accrued interest and certain short-term instruments may use simple interest — read the problem wording carefully.
Pitfall 3: Sign Convention
On the BA II Plus, PV and FV must have opposite signs. If PV and FV have the same sign → your cash flow direction is set incorrectly.
Pitfall 4: PV ≠ "Original Principal"
PV is not the amount you invested. PV is the "equivalent amount today" of a future cash flow — it's a calculated result. Only when you compare the PV of future returns against the initial outlay (as in Example 4's NPV approach) does the comparison make sense.
7. Core Formula Cheat Sheet
| Condition | Formula | Notes |
|---|---|---|
| Compound PV (Standard) | PV = FV / (1+r)ⁿ | 🔥 Most commonly used |
| Compound PV (Alternative) | PV = FV × (1+r)⁻ⁿ | Discount factor form |
| Simple PV | PV = FV / (1+r×n) | Use only when explicitly stated |
| Non-annual compounding | r_per = r/m, n_per = n×m | m = compounding periods per year |
📝 Practice Questions
Q1 (Basic Concept)
You deposit a lump sum today at 8% annual compound interest and receive $50,000 after 5 years. How much must you deposit today? (Round to nearest dollar)
A. $30,000 B. $34,029 C. $36,050 D. $40,000
Q2 (Discount Rate Comparison)
For a $10,000 cash flow 10 years from now, under which discount rate is the PV highest?
A. r = 5% B. r = 8% C. r = 10% D. r = 12%
Q3 (Compounding Frequency Trap)
A future cash flow FV = $20,000, n = 2 years, annual rate 10%, compounded quarterly. The PV is closest to:
A. $16,200 B. $16,400 C. $16,529 D. $17,000
Q4 (NPV Decision)
A project requires an investment of $50,000 today and returns a single payment of $65,000 in 3 years. Your required rate of return is 10%. Which statement is correct?
A. NPV > 0, should invest B. NPV < 0, should not invest C. NPV = 0, indifferent D. Insufficient information to decide
Q5 (Simple vs. Compound)
Same FV = $10,000, n = 4 years, r = 5%. The PV under simple interest discounting compared to compound interest discounting is:
A. Higher B. Lower C. Equal D. Cannot be determined
📋 Answers
| Q# | Answer | Explanation |
|---|---|---|
| Q1 | B | PV = 50,000 / (1.08)⁵ = 50,000 / 1.4693 = $34,029 |
| Q2 | A | Lower discount rate → higher PV. r = 5% gives the largest discount factor |
| Q3 | C | r_per = 10%/4 = 2.5%, n = 8 → PV = 20,000 / (1.025)⁸ = 20,000 / 1.2184 = $16,529 |
| Q4 | B | PV = 65,000 / 1.1³ = 65,000 / 1.331 = $48,835 < $50,000 → NPV = -$1,165 < 0 |
| Q5 | A | Simple discount factor = 1/(1+0.05×4) = 0.8333; Compound = 1/(1.05)⁴ = 0.8227 → Simple PV is higher |
🎯 Master PV calculation, and you hold the "discounting" key to the TVM universe. Next lesson L090 covers FV calculation (single cash flow) — once you're fluent in both directions, you'll move freely through time.