Interest Rates - Detailed Notes

Interest Rates - Detailed Notes

Key Concepts of Interest Rates

  • Interest Rates: The price of using money; it represents the cost of borrowing or the return on investment.
  • Types of Interest Rate Quotes:
  • Effective Annual Rate (EAR): Total interest earned on an investment after one year of compounding.
    • Example: An investment of $100 at an EAR of 5% grows as follows:
    • 6 months: $102.47 = $100 × (1.05)^0.5
    • 1 year: $105.00 = $100 × (1.05)^1.0
    • 2 years: $110.25 = $100 × (1.05)^2.0
  • Annual Percentage Rate (APR): Indicates the annual interest without compounding effects; does not reflect the actual amount earned when compounding occurs.

Interest Rate Adjustments

  • Adjusting the discount rate to the cash flow time period is crucial for accurate Present Value (PV) and Future Value (FV) calculations.
  • Equivalent n-Period Discount Rate formula:
  • ( 1 + r = (1 + r)^{n} - 1 ) (Eq. 3.1)

Example: Valuing Monthly Cash Flows

  • Scenario: Saving to accumulate $100,000 in 10 years with a bank account that offers a 6% EAR.

  • Step 1: Calculate the monthly discount rate:

  • Equivalent monthly discount rate: 0.4868% per month.

  • Step 2: Identify relevant valuation formula for the future value of an annuity:

  • ( FV = C \times \frac{(1 + r)^n - 1}{r} )

  • Step 3: Solve for monthly contribution (C):

  • ( C = \frac{FV \times r}{(1 + r)^n - 1} ) leading to:

  • Monthly Contribution (C): Approximately $615.47 per month.

Converting APR to EAR

  • To find the effective rate, the APR can be converted to EAR using the formula:
  • ( EAR = (1 + \frac{APR}{m})^m - 1 ) (Eq. 3.3)
  • Where m is the number of compounding periods per year.

Amortizing Loans

  • An Amortizing Loan is repaid through scheduled payments that include interest plus principal.
  • Example Questions:
  1. Loan Payment Calculation: For a $30,000 loan at 6.75% APR over 60 months, the monthly payment is approximately $590.08.
  2. Outstanding Balance Calculation: After three years into a $30,000 loan (6.75% APR), the remaining balance after 36 months is approximately $13,222.32.

Determinants of Interest Rates

  • Interest rates are influenced by supply and demand for funds:

  • Components of Interest Rate:

    • Real risk-free interest rate
    • Inflation premium
    • Risk premium

  • Nominal vs. Real Rates:

  • Nominal Rate: Rate without adjustments for inflation.

  • Real Rate: Rate adjusted for inflation, indicating growth in purchasing power.

  • Fisher Equation describes the relationship:

    • ( 1 + r = (1 + r_{real})(1 + \pi) )

  • Example Real Interest Rate Calculation:

  • 2005: Nominal rate of 5.1%, inflation 2.7% yields a real rate of approximately 2.34%.

  • 2016: Nominal rate of 1.7%, inflation 1.3% yields a real rate of approximately 0.39%.