14 Chapter 3: Time Value of Money

Conceptual Questions

1. Why is Money Today Worth More?

Money today is worth more because:

•Opportunity Cost: It can be invested to earn returns.

•Inflation: Over time, inflation decreases the purchasing power of money.

•Risk: Receiving money in the future involves uncertainty.

2. Future Value and Compounding

Compounding leads to exponential growth because interest is earned not only on the original principal but also on previously earned interest. The more frequent the compounding, the greater the future value due to the accumulation of interest on a shorter time scale.

3. Present Value and Discounting

Increasing the discount rate reflects a higher opportunity cost or risk. This decreases the present value because future cash flows are “discounted” more heavily, making them worth less today.

4. Applications of TVM

•Personal Finance Example: Saving for retirement using compound interest.

•Business Finance Example: Evaluating the present value of future cash flows for a project.

Short Calculations

1. Simple Future Value Calculation

FV = PV × (1 + r)^n

FV = $2,000 × (1 + 0.06)^4 = $2,000 × 1.2625 = $2,525

2. Present Value of a Future Sum

PV = FV ÷ (1 + r)^n

PV = $5,000 ÷ (1 + 0.08)^10 = $5,000 ÷ 2.1589 ≈ $2,315.24

3. Future Value with Non-Annual Compounding

FV = PV × (1 + r/m)^(n × m)

FV = $3,000 × (1 + 0.05/12)^(6 × 12)

FV = $3,000 × (1.004167)^72 ≈ $3,000 × 1.34885 ≈ $4,046.55

4. Effective Annual Rate (EAR)

EAR = (1 + r/m)^m − 1

EAR = (1 + 0.10/4)^4 − 1 = (1.025)^4 − 1 ≈ 10.38%

Scenario-Based Problems

1. Comparing Investment Options

•Option A: FV = $10,000 × (1 + 0.05)^8 = $10,000 × 1.47746 ≈ $14,774.60

•Option B: FV = $10,000 × (1 + 0.048/12)^(8 × 12)

FV = $10,000 × (1.004)^96 ≈ $10,000 × 1.43236 ≈ $14,323.60

•Conclusion: Option A provides the higher future value.

2. Retirement Planning

FV = PMT × [(1 + r/m)^(n × m) − 1] ÷ (r/m)

FV = $500 × [(1 + 0.07/12)^(30 × 12) − 1] ÷ (0.07/12)

FV = $500 × [(1.005833)^360 − 1] ÷ 0.005833

FV ≈ $500 × [6.022575 − 1] ÷ 0.005833 ≈ $500 × 861.11 ≈ $430,555.00

3. Loan Decisions

•Monthly Payment: PMT = PV × [r/m ÷ (1 − (1 + r/m)^−(n × m))]

PMT = $50,000 × [0.06/12 ÷ (1 − (1 + 0.06/12)^−60)]

PMT = $50,000 × [0.005 ÷ (1 − (1.005)^−60)] ≈ $50,000 × 0.01933 ≈ $966.50

•Total Interest Paid = (PMT × Total Payments) − Loan Amount

Total Interest Paid = ($966.50 × 60) − $50,000 = $57,990 − $50,000 ≈ $7,990

•EAR = (1 + r/m)^m − 1

EAR = (1 + 0.06/12)^12 − 1 ≈ 6.17%

Case Study: Lump Sum vs. Annuity

Scenario

• • Option A: $500,000 today.
  • Option B: $50,000 annually for 15 years.
  • Discount rate: 6%.

Tasks and Solutions

1. 1. Calculate the Present Value (PV) of Option B:

The PV of an annuity can be calculated using the formula:

[latex]PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}[/latex]

Where:

• • PMT = 50,000 (annual payment)
  • r = 0.06 (annual discount rate)
  • n = 15 (number of payments)

Step-by-Step Calculation:

[latex]PV = 50,000 \times \frac{1 - (1 + 0.06)^{-15}}{0.06}[/latex]

• • First, calculate (1 + 0.06)^{-15} :

[latex](1 + 0.06)^{-15} = (1.06)^{-15} \approx 0.417265[/latex]

• • Subtract from 1:

1 – 0.417265 = 0.582735

• • Divide by r = 0.06 :

[latex]\frac{0.582735}{0.06} \approx 9.71225[/latex]

• • Multiply by PMT = 50,000 :

[latex]PV = 50,000 \times 9.71225 \approx 485,612.50[/latex]

Result: The Present Value of the annuity (Option B) is $485,612.50.

2. Which option is better?

• • Option A: $500,000 today.
  • Option B: $485,612.50 (calculated PV).

Based on the calculation:

• • Option A provides a slightly higher value of $500,000.
  • Option B’s value is lower because of the discounting effect over 15 years.

Conclusion:

From a purely financial perspective, Option A (lump sum of $500,000 today) is the better choice, assuming no other factors such as investment risk or personal spending preferences.

Interactive Challenge

1. Option 1: Lump Sum Investment

FV = PV × (1 + r)^n

FV = $75,000 × (1 + 0.04)^8 ≈ $75,000 × 1.36857 ≈ $102,643

2. Option 2: Monthly Contributions

FV = PMT × [(1 + r/m)^(n × m) − 1] ÷ (r/m)

FV = $800 × [(1 + 0.06/12)^(8 × 12) − 1] ÷ (0.06/12)

FV ≈ $800 × [1.85093 − 1] ÷ 0.005 ≈ $800 × 170.186 ≈ $136,149

Conclusion: The monthly contribution option meets the $100,000 goal and provides more value.