Chapter 5 - Interest Rates - In-depth Notes

5.1 What is the interest rate?

  • Definition: Interest rates represent the price of using money.
    • It functions as the rental price for money.
    • For borrowers, it denotes the cost of early consumption before earning.
    • For savers, it is the reward for postponing consumption.

5.1 Interest Rate Quotes and Adjustments

  • Learning Objective: Understand how interest rates can be quoted in various ways.

5.1 Interest Rate Quotes

  • Compounding Periods: Interest rates can differ based on compounding periods, such as:

    • Annual
    • Semi-annual
    • Quarterly
    • Monthly
    • Daily
    • Continuous

  • Expressing Interest Rates:

    • Annual Percentage Rate (APR): The stated interest rate with no adjustments for compounding.
    • Effective Annual Rate (EAR): Reflects the actual interest earned or paid after accounting for compounding within a year.
    • Periodic Rate: The interest rate applied for a specific compounding period.

5.1 APR Details

  • APR Calculation:

    • APR is not used directly in Time Value of Money (TVM) calculations.
    • To find the periodic rate: \text{Periodic Rate} = \frac{\text{APR}}{m} where $m$ is the number of compounding periods per year.

  • Example: For a 6% APR compounded monthly:

    • \text{Periodic Rate} = \frac{0.06}{12} = 0.005 \text{ or } 0.5\%

  • Limitations: APR does not represent the actual rate earned due to compounding effects.

5.1 Effective Annual Rate (EAR)

  • Definition: EAR shows the true amount of interest earned in one year, accounting for compounding.
  • Usage: Comparisons of investments or loans that have different compounding frequencies.

5.1 Conversions Between APR and EAR

  • Converting APR to EAR And Vice Versa:
    • Convert an APR to an EAR using: \text{EAR} = \left(1 + \frac{\text{APR}}{m}\right)^{m} - 1
    • Convert EAR to APR using: \text{APR} = \left(1 + \text{EAR}\right)^{\frac{1}{m}} - 1

5.2 Application: Discount Rates and Loans

  • Purpose: Use quoted interest rates to calculate loan payments and outstanding loan balances.

  • Major Types of Loans:

    1. Pure Discount Loan (Zero Coupon Bond)
    2. Interest-Only Loan
    3. Amortized Loan

5.2 Loan Amortization

  • Each payment includes both interest and principal reductions.

  • Interest Calculation: \text{Interest Paid} = \text{Beginning Balance} \times \text{Interest Rate}

  • Principal Paid: \text{Principal Paid} = \text{Total Payment} - \text{Interest Paid}

  • At Loan End: Entire principal must be paid off.

5.3 The Determinants of Interest Rates

  • Learning Objective: Understand inflation, expectations, and risk's role in setting interest rates.

Supply and Demand for Funds

  • Interest rates are set by market forces where supply meets demand.
  • Factors influencing demand:
    • Production opportunities
    • Technological advances
    • Tax policy changes

Inflation and Real vs. Nominal Rates

  • Nominal Interest Rate: Rate before adjusting for inflation.
  • Real Interest Rate: Rate adjusting for inflation impact on purchasing power.
    • Formula: 1 + r = \frac{1 + i}{1 + \text{Inflation}}

Yield Curve

  • Relationship between bond yields and maturities.
    • Normal Yield Curve: Longer-term rates are higher.
    • Inverted Yield Curve: Short-term rates exceed long-term rates, indicating expected economic slowdowns.

5.4 The Opportunity Cost of Capital

  • Definition: The expected return for investing in an alternative with similar risk.
  • Example: If given an option to lend $100 with an expected return lower than available market rates, you may choose not to lend.

Chapter Quiz and Formulas

  • Differences between EAR and APR.
  • Calculating changing loan interest components over time.
  • The implications of nominal vs. real interest rates in decision-making.