Time Value of Money (TVM)

Core Concept

The Fundamental Principle

A dollar today is worth more than a dollar in the future

Why? Because time is money - you can invest today’s dollar to earn returns.

TVM Definition

Adjusts for the fact that dollars received/paid in the future are not equivalent to those received/paid today
Most important concept in personal finance

Two Main Components

Future Value (FV): What an investment will be worth in the future
Present Value (PV): What a future amount is worth today

Interest Calculation Methods

Simple Interest

Formula:

i = p × r × t = prt

Where:

p = principal (amount invested)
r = interest rate (as decimal)
t = time in years

Example:

Principal: $1,000
Rate: 8% (0.08)
Time: 4 years
Interest: $1,000 × 0.08 × 4 = $320

Limitation: Assumes interest is withdrawn each year; only principal earns interest

Compound Interest

Definition

Earning interest on interest
The best way to build investment values over time
Always assumed in TVM calculations

How It Works

Example: $1,000 at 8% for 4 years

Year	Starting Amount	Interest (8%)	Ending Amount
1	$1,000.00	$80.00	$1,080.00
2	$1,080.00	$86.40	$1,166.40
3	$1,166.40	$93.31	$1,259.71
4	$1,259.71	$100.78	$1,360.49

Total Interest: $360.49 (vs. $320 with simple interest)

Deriving the Formula

Year 1:

V₁ = p + pr = p(1 + r)

Year 2:

V₂ = V₁(1 + r) = p(1 + r)(1 + r) = p(1 + r)²

Year 3:

V₃ = V₂(1 + r) = p(1 + r)²(1 + r) = p(1 + r)³

General Formula:

Vₜ = p(1 + r)ᵗ

Compound Interest Formula

Compound Interest = p[(1 + r)ᵗ - 1]

Future Value (FV)

Definition

Valuation of an asset projected to the end of a time period in the future
What your investment will be worth

Formula

FV = PV(1 + r)ᵗ

Where:

PV = Present Value (current amount)
r = interest rate per period
t = number of time periods

Example

Invest $1,000 at 8% for 4 years
FV = $1,000(1 + 0.08)⁴ = $1,360.49

The Rule of 72

Quick Doubling Calculator

Formula:

Years to double = 72 ÷ interest rate

Examples:

7% interest: 72 ÷ 7 = 10.3 years to double
6% interest: 72 ÷ 6 = 12 years to double
8% interest: 72 ÷ 8 = 9 years to double

Use: Quick mental calculation for investment growth

Future Value of Annuity (FVA)

Definition

Annuity: Series of equal payments made at regular intervals
FVA: What those payments will be worth in the future

Formula

FVA = PMT × [(1 + r)ᵗ - 1] / r

Where:

PMT = payment amount per period
r = interest rate per period
t = number of periods

Example

Save $1,000 per year for 10 years
Earn 10% per year
FVA = $1,000 × [(1 + 0.1)¹⁰ - 1] / 0.1 = $15,937.42

Present Value (PV)

Definition

Current value of an asset (or stream of assets) to be received in the future
Amount needed today to reach a future goal

Formula

PV = FV / (1 + r)ᵗ

Alternative notation:

PV = FV(1 + r)⁻ᵗ

Example

Want $20,000 in 10 years
Can earn 7% return
PV = $20,000 / (1 + 0.07)¹⁰ = $10,167

Interpretation: Need to invest $10,167 today to have $20,000 in 10 years at 7%

Present Value of Annuity (PVA)

Definition

Current worth of a stream of future payments
Useful for retirement planning

Formula

PVA = PMT × [1 - 1/(1 + r)ᵗ] / r

Example: Retirement Planning

Want $30,000 per year for 20 years in retirement
Can earn 7% return
PVA = $30,000 × [1 - 1/(1 + 0.07)²⁰] / 0.07 = $317,820

Interpretation: Need $317,820 at retirement (not $600,000!) to fund $30,000/year for 20 years at 7%

Power of Compounding

Comparison: 40 Years at 8%

Simple Interest:

i = $1,000 × 0.08 × 40 = $3,200

Compound Interest:

i = $1,000[(1 + 0.08)⁴⁰ - 1] = $20,724.52

Difference: $17,524.52 more with compounding!

Key Insight

“The way to build wealth is to make money on your money, not simply to put money away”

Key Formulas Summary

Concept	Formula	Use
Simple Interest	`i = prt`	Basic interest calculation
Future Value	`FV = PV(1 + r)ᵗ`	What investment will be worth
Present Value	`PV = FV/(1 + r)ᵗ`	What to invest today
Rule of 72	`Years = 72/rate`	Quick doubling time
FV Annuity	`FVA = PMT[(1+r)ᵗ-1]/r`	Future value of payments
PV Annuity	`PVA = PMT[1-1/(1+r)ᵗ]/r`	Present value of payments

Practical Applications

1. Savings Goals

Use FV to see what regular savings will become
Use PV to determine how much to save now

2. Retirement Planning

Use PVA to calculate retirement nest egg needed
Use FVA to see what regular contributions will accumulate to

3. Investment Decisions

Compare different investment options using TVM
Evaluate whether returns justify the investment

4. Loan Analysis

Calculate total cost of borrowing
Compare loan options

Exam Tips

✅ Master the formulas - they’re essential
✅ Know when to use FV vs. PV
✅ Understand annuities (series of payments)
✅ Remember: compound interest > simple interest
✅ Rule of 72 is for quick mental calculations
✅ PV and FV are inversely related
✅ Always convert percentages to decimals (8% = 0.08)
✅ Time periods and interest rate must match (annual rate = annual periods)

Common Mistakes to Avoid

❌ Using percentage instead of decimal (use 0.08, not 8)
❌ Mixing time periods (annual rate with monthly periods)
❌ Forgetting to subtract 1 in compound interest formula
❌ Confusing PV and FV
❌ Using simple interest when compound is appropriate
❌ Forgetting parentheses in calculations

Practice Problem Types

Type 1: Future Value

“How much will $X grow to in Y years at Z%?”

Use: FV = PV(1 + r)ᵗ

Type 2: Present Value

“How much do I need today to have $X in Y years at Z%?”

Use: PV = FV/(1 + r)ᵗ

Type 3: Future Value Annuity

“If I save $X per year for Y years at Z%, how much will I have?”

Use: FVA = PMT[(1+r)ᵗ-1]/r

Type 4: Present Value Annuity

“How much do I need to fund $X per year for Y years at Z%?”

Use: PVA = PMT[1-1/(1+r)ᵗ]/r

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