15

Conceptual Questions

• • TVM means money today is worth more than the same amount in the future due to earning potential, inflation, and opportunity costs.

2.

• • Principal invested (more money = more future value).
  • Interest rate (higher rates = faster growth).
  • Time (longer time = more compounding periods).

3.

• • Simple interest: Only principal earns interest.
  • Compound interest: Interest earns interest.
    Example: A savings account growing faster over years through compounding.

4.

• • More frequent compounding = faster growth.
  • Annual compounding grows slower than monthly or daily compounding.

5.

• • Higher discount rates lower Present Value.
  • They reflect greater opportunity cost or investment risk.

6.

• • Rule of 72 estimates doubling time: 72 ÷ interest rate.
  • Less accurate at very high or very low rates.

7.

• • Ordinary Annuity: Payments at end of each period (e.g., loans).
  • Annuity Due: Payments at beginning (e.g., rent).
  • Annuity Due grows faster.

8.

• • Amortization = fixed payments split between interest and principal.
  • Early payments mainly pay interest, later ones pay principal.

9.

• • Early saving uses compounding to maximize growth.
  • Waiting costs future value.

10.

Apps and calculators speed up solving but understanding TVM is essential for proper decision-making.

Problem Solving

1.
FV = 3,000 × (1 + 0.07)^10 ≈ $5,904.71

2.
FV = 8,000 × (1 + 0.06/12)^(12×5) ≈ $10,745.29

3.
PV = 20,000 ÷ (1 + 0.05)^6 ≈ $14,925.68

4.
72 ÷ 9 = 8 years

5.

• • Investment A EAR = 5%
  • Investment B EAR = (1 + 0.048/12)^12 – 1 ≈ 4.91%
    Investment A has a slightly higher effective rate.

6.
FV = 400 × [(1 + 0.06/12)^(12×25) – 1] ÷ (0.06/12) ≈ $279,072.22

7.
PV = 50,000 × [(1 – (1 + 0.05)^-20) ÷ 0.05] ≈ $623,110.45

8.
PMT = 15,000 × [0.005(1 + 0.005)^60] ÷ [(1 + 0.005)^60 – 1] ≈ $289.88

9.
Paying early reduces principal, shortens loan term, and reduces total interest paid.

10.

• • PV of $12,000 = 12,000 ÷ (1 + 0.05)^5 ≈ $9,415.

$10,000 today > $9,415 later.
Option A is better.

Interactive Challenge

1. 1. Quick Decision

• • 6% → 12 years
  • 8% → 9 years
  • 12% → 6 years

1. 2. Choose Best Compounding Option

• • 6% compounded daily gives the highest future value.

1. 3. Identify the Mistake

• • Mistake: Believing all payments reduce principal equally.
  • In reality, early payments are mostly interest.

1. 4. Lump Sum vs. Stream of Payments

• • Likely the lump sum of $10,000 is better when discounted at 5%.

1. 5. Savings Plan Adjustment

• • Increasing monthly savings boosts final balance significantly due to new contributions compounding over time.

1. 6. Strategy Challenge

• • Applying the bonus toward principal saves much more in interest over time versus keeping it at low savings account returns.

1. 7. Timeline Building

• • Monthly compounding curve grows faster and curves upward more steeply than annual.

1. 8. Future Value Estimation

• • 72 ÷ 9 = 8 years (approximately).