📖 货币时间价值导论 — PV 与 FV 概念

📌 Topic: Introduction to Time Value of Money — Present Value (PV) and Future Value (FV) Concepts

1. Review & Introduction

In the previous six lessons (L081–L086), we completed the Probability & Probability Distributions module — from random variables, PDF/CDF, normal distribution, to t-distribution and the Central Limit Theorem. We built a toolkit for describing uncertainty in the language of probability.

Starting from L087, we enter a new module: Time Value of Money (TVM). This module carries the largest weight in CFA Level 1 Quantitative Methods (12 lessons, L087–L098) and is the most fundamental and universally applicable computational skill across all CFA levels — underpinning equity valuation, bond pricing, capital budgeting, derivatives pricing, and nearly every other domain.

This lesson is the starting point of TVM. We cover the two core concepts: Present Value (PV) and Future Value (FV), along with the bridge that connects them — the interest rate.

2. Core Concept Explanations

2.1 What Is the Time Value of Money?

The core idea of Time Value of Money (TVM) is captured in a single sentence:

🔑 A dollar today is worth more than a dollar tomorrow.

This is not a platitude — it is a mathematical fact. Three reasons:

1. Opportunity Cost: Money today can be invested immediately and earn interest by tomorrow
2. Inflation: Rising prices erode the purchasing power of future money
3. Uncertainty: Future receipts carry risks of default, delay, etc.

Mathematical Expression of TVM:

$$FV = PV \times (1 + r)^n$$

$$PV = \frac{FV}{(1 + r)^n}$$

Where: - r = interest rate per period (discount rate) - n = number of periods - FV = Future Value (value at a future point in time) - PV = Present Value (value at the current point in time)

2.2 Future Value (FV) — "How much will today's money be worth tomorrow?"

Future Value (FV): The future value of a sum of money after a given number of periods at a given interest rate.

Core Intuition: Deposit money in a bank, compound interest, it grows and grows.

FV Formula (single cash flow, compound interest):

$$FV = PV \times (1 + r)^n$$

Example 1: Simple FV Calculation

You deposit 10,000 at an annual interest rate of 5% for 3 years. What is the total amount at maturity?

Solution:

$$FV = 10{,}000 \times (1 + 0.05)^3$$ $$FV = 10{,}000 \times 1.157625$$ $$FV = 11{,}576.25$$

Answer: 11,576.25

Over three years the investment grew by 1,576.25, approximately 15.76% — that is the value of time.

Example 2: The Exponential Effect in FV

100 at an annual rate of 10%, observe the FV over different horizons:

Key Observation: The longer the horizon, the more dramatic the compounding effect — the first 10 years multiply the principal by ~2.6×, and the next 10 years multiply it again by ~2.6×. This is the "magic of compounding."

2.3 Present Value (PV) — "How much is tomorrow's money worth today?"

Present Value (PV): A future cash flow, discounted back to today at a given discount rate.

Core Intuition: "Discounting" future money back to the present.

PV Formula (single cash flow):

$$PV = \frac{FV}{(1 + r)^n}$$

Here, r is called the discount rate.

Example 3: Simple PV Calculation

You will receive 10,000 in 3 years. The market interest rate is 5% (annual compounding). What is this worth today?

Solution:

$$PV = \frac{10{,}000}{(1 + 0.05)^3}$$ $$PV = \frac{10{,}000}{1.157625}$$ $$PV = 8{,}638.38$$

Answer: 8,638.38

Interpretation: If you had 8,638.38 today and invested it at 5%, it would grow to exactly 10,000 in 3 years. Thus, 10,000 in the future is "equivalent" to 8,638.38 today.

Example 4: The "Discounting" Nature of PV

Using the reverse perspective of Example 2 — 100 receivable at various future dates, discounted at 10%:

Key Observation: The further in the future, the lower the present value. 100 receivable in 30 years is worth only 5.73 today — illustrating how severely distant money loses value.

2.4 The Symmetric Relationship Between PV and FV

PV and FV are two sides of the same coin:

Core Formula Comparison:

$$FV = PV \times \underbrace{(1+r)^n}_{\text{FV Factor}}$$

$$PV = FV \times \underbrace{\frac{1}{(1+r)^n}}_{\text{PV Factor}}$$

3. Deeper Discussion

3.1 The Three Roles of the Interest Rate

Along the same timeline, the interest rate r plays three roles:

All three are fundamentally the same: "the price of time for money."

CFA exam questions often use these three terms interchangeably — you need to interpret from context — but mathematically they all use the same formula.

3.2 The Timeline — The Essential TVM Tool

Drawing a timeline is the single most important habit for solving TVM problems. Standard format:

t=0      t=1      t=2      t=3 ...   t=n
|--------|--------|--------|---------|
PV                 CF1      CF2      FV

• t=0 = today (present)
• t=n = end of period
• Arrow direction: PV → FV (compounding), FV → PV (discounting)

Example 5: Using a Timeline to Assist Understanding

You plan to buy a car in 5 years, expecting to need 200,000. You currently have 150,000 to invest. What annual return do you need to achieve your goal?

t=0          t=5
|------------|
150,000     200,000

PV = 150,000  →  FV = 200,000
Find r:

$$200{,}000 = 150{,}000 \times (1 + r)^5$$ $$(1 + r)^5 = \frac{200{,}000}{150{,}000} = 1.3333$$ $$1 + r = 1.3333^{1/5} = 1.0592$$ $$r = 5.92\%$$

Answer: Annualized required return ≈ 5.92%

This "given PV and FV, solve for r" question type is extremely common in the CFA exam.

3.3 Relationship Between FV Factor and PV Factor

$$FV\ Factor = (1 + r)^n$$ $$PV\ Factor = \frac{1}{(1 + r)^n} = (1 + r)^{-n}$$

The two are reciprocals of each other. Exam questions often give you the PV Factor and ask you to derive the FV Factor, or vice versa.

4. Key Methods & Tools

4.1 Calculator Operation Primer

Texas Instruments BA II Plus (CFA-approved calculator):

Computing FV (given PV): | Keystroke | Description | |-----------|-------------| | 10,000 PV | Enter present value | | 5 I/Y | Enter annual interest rate | | 3 N | Enter number of periods | | 0 PMT | No annuity payment | | CPT FV | Compute future value | | Display -11,576.25 | ✅ |

Computing PV (given FV): | Keystroke | Description | |-----------|-------------| | 10,000 FV | Enter future value | | 5 I/Y | Enter discount rate | | 3 N | Enter number of periods | | 0 PMT | No annuity payment | | CPT PV | Compute present value | | Display -8,638.38 | ✅ |

⚠️ On the BA II Plus, PV and FV will have opposite signs (one positive, one negative) because they represent opposite directions of cash flow (inflow vs. outflow).

4.2 Manual Calculation Methods (Exam Edge)

When rates are simple and periods few, manual calculation can be faster than the calculator:

• Approximating (1 + r)^n: when r is small and n is not large, (1+r)^n ≈ 1 + n×r (first-order approximation)
• Compound factor tables: The exam may provide PV Factor / FV Factor look-up tables

5. Practical Insights & Cautions

🚩 Common Pitfalls

1. Confusing the direction of PV and FV: PV→FV is multiplication by (1+r)^n; FV→PV is division by (1+r)^n. Drawing a timeline solves all
2. Mismatched rate and period: Annual rate pairs with years, monthly rate pairs with months — r and n must be in the same time dimension
3. Forgetting the sign convention: On the calculator, PV and FV have opposite signs (inflow vs. outflow)
4. Treating compound interest as simple periodic interest: Compound interest "earns interest on interest" — it is not a fixed amount per period
5. Assuming PV is always less than FV: If the interest rate is negative (rare), PV could exceed FV

💡 Memory Aids

• "PV divide, FV multiply" — PV = FV ÷ (1+r)^n, FV = PV × (1+r)^n
• Discounting is reverse compounding — compounding grows today's money into the future; discounting shrinks future money back to today
• The interest rate is the price of time — banks pay interest because they are essentially "renting" your time

6. Key Formula Card for This Lesson

📝 Practice Questions

Q1

You deposit 5,000 in a bank today at an annual interest rate of 4% (annual compounding). The account balance after 5 years is closest to:

A. 6,000 B. 6,083 C. 6,500 D. 7,000

Q2

A sum of 50,000 receivable in 8 years is discounted to today at an annual discount rate of 6%. Its present value is closest to:

A. 31,000 B. 31,370 C. 35,000 D. 42,000

Q3

Regarding the time value of money, which of the following statements is incorrect?

A. The future value factor and the present value factor are reciprocals of each other B. If the interest rate is positive, the present value is always less than the future value C. The higher the discount rate, the lower the present value of a future cash flow D. For a given FV and n, a higher interest rate r results in a lower PV

Q4

An investor wishes to accumulate 200,000 in 10 years. If the annual return is 7% (annual compounding), how much principal must be invested as a lump sum today?

A. 93,458 B. 101,670 C. 105,340 D. 112,290

Q5

For a single cash flow, with r = 8% and n = 9, what is the product of the future value factor (1+r)^n and the present value factor 1/(1+r)^n?

A. 0 B. 0.5 C. 1.0 D. 2.0

✅ Answers & Explanations

🔜 模块二最后一课。更多数量分析课程后续更新。