Chapter 5

Key Concepts and Skills

  • Future Value of Multiple Cash Flows
    • Ability to compute the future value of multiple cash flows.
    • Example: Depositing $4,000 at the end of each year for 3 years with an 8% interest rate.
  • Present Value of Multiple Cash Flows
    • Ability to compute the present value of multiple cash flows.
    • Applying the time value of money (TVM) principle in various financial scenarios.
  • Loan Payments
    • Ability to compute payments for loans.
    • Understanding of loan amortization and various loan types.
  • Interest Rate Calculation
    • Ability to find the interest rate on a loan.
  • Amortization of Loans
    • Understanding how loans are amortized or paid off.
  • Interest Rate Quotations
    • Understanding how interest rates are quoted and compared across different loans.

Future and Present Values of Multiple Cash Flows

  • Chapter Sections
    • 5.1 Future and Present Values of Multiple Cash Flows.
    • 5.2 Valuing Level Cash Flows: Annuities and Perpetuities.
    • 5.3 Comparing Rates: The Effect of Compounding Periods.
    • 5.4 Loan Types and Loan Amortization.

Examples of Future Value Calculations

  • Example 1: Future Value of Cash Flows
    • Initial deposit = $7,000.
    • Annual deposits = $4,000 for 3 years at 8% interest.
    • Total Future Value calculations:
    • Year 0: FV = $7,000 × (1.08)³ = $8,817.98
    • Year 1: FV = $4,000 × (1.08)² = $4,665.60
    • Year 2: FV = $4,000 × (1.08)¹ = $4,320.00
    • Year 3: FV = $4,000.00
    • Total value in 3 years = $21,803.58
  • Example 2: Future Value with Different Cash Inflows
    • If you deposit $100, $200, and $300 at different years:
    • Calculate total FV at 3 years and 5 years at 7% interest.

Present Value Calculations

  • Example 1: Present Value of Cash Inflows
    • Payments of $200, $400, $600, and $800 over 4 years with a 12% return.
    • Total present value calculations will include discounting each future payment back to present value.
  • Example 2: Present Value of Future Payments
    • Calculation method to determine how much you should pay for certain cash flows.

Annuities and Perpetuities

  • Key Definitions
    • Annuity: Series of equal payments at regular intervals.
    • Ordinary Annuity: Payments occur at the end of each period.
    • Annuity Due: Payments occur at the beginning of each period.
    • Perpetuity: Infinite series of equal payments.
  • Formula Examples
    • PV of Annuity: PMTr(1−1(1+r)t)\frac{PMT}{r} \left(1 - \frac{1}{(1 + r)^t}\right)
    • PV of Perpetuity: PMTr\frac{PMT}{r}

Loan Types and Amortization

  • Types of Loans
    • Pure Discount Loans: Loans that do not require interest payments until maturity (e.g., T-bills).
    • Interest-Only Loans: Only interest is paid until the principal is repaid at the end.
    • Amortized Loans: Regular payments cover interest and reduce principal over time, common in consumer loans.
  • Loan Amortization
    • Calculation of total payments, interest, principal paid, and balance over time using an amortization schedule.

Interest Rates

  • APR (Annual Percentage Rate)
    • Refers to the annualized interest rate including fees. Calculated as APR=r∗mAPR = r * m where (m) is the number of compounding periods.
  • EAR (Effective Annual Rate)
    • Adjusts APR for compounding. Calculated as EAR=(1+APRm)m−1EAR = (1 + \frac{APR}{m})^m - 1

Important Notes

  • Use APR for comparing loans when rates are quoted irregularly (daily, monthly, etc.).
  • Always adjust rates based on compounding periods when making financial calculations.