Detailed Study Notes on Interest Rates

Interest Rates Overview
- Interest rates serve as the price of using money.
Effective Annual Rate (EAR)
- Definition: The total amount of interest that will be earned at the end of one year.
- Example Calculations:
- With an EAR of 5%, a $100 investment grows as follows:
  - After 6 months:
    100 imes (1.05)^{0.5} = 102.47
  - After 1 year:
    100 imes (1.05)^{1.0} = 105.00
  - After 2 years:
    100 imes (1.05)^{2.0} = 110.25
Adjusting Discount Rate to Different Time Periods
- When computing Present Value (PV) or Future Value (FV), the discount rate must align with the cash flows' time period.
- Utilize the formula for Equivalent n-Period Discount Rate:
  \text{Equivalent n-Period Discount Rate} = (1 + r)^{n} - 1

Example: Savings Accumulation
- Problem Statement: Determine the required monthly savings to accumulate $100,000 over 10 years with a bank account that has an EAR of 6%.
Step 1: Calculate Equivalent Monthly Discount Rate
- Given the EAR of 6%, find the monthly rate:
  r = 6\% \rightarrow \frac{r}{12} = 0.4868\% \text{ per month}
Step 2: Determine Inputs and Valuation Formulas
- Inputs:
- Future Value (FV) = $100,000
- Number of periods (n) = 120 months
- Monthly interest rate = 0.4868%
- Future Value of Annuity Formula:
  FV = C \times \left( \frac{(1 + r)^{n} - 1}{r} \right)
Step 3: Solve for Monthly Savings (C)
- Rearranging the formula to find C:
  C = \frac{FV}{\left(\frac{(1 + r)^{n} - 1}{r}\right)}
- Calculation:
  C = \frac{100,000}{\left(\frac{(1.004868)^{120} - 1}{0.004868}\right)} \approx 615.47

Annual Percentage Rates (APR)
- Definition: The amount of interest earned in one year, not accounting for compounding effects.
- Limitations:
- Does not reflect actual earnings over the year.
- Cannot be used as a discount rate.
- Formula for interest rate per compounding period:
  \text{Interest rate per compounding period} = \frac{APR}{m} \quad (m = \text{number of compounding periods per year})
Converting APR to EAR
- EAR is calculated from APR using:
  EAR = (1 + \frac{APR}{m})^{m} - 1
- Where m is the number of compounding periods per year.
Lecture Example (EAR for 6% APR)
- Changes in compounding period affect EAR calculations.

Common Financial Problems
- Calculate required loan repayments.
- Determine the remaining balance on an amortizing loan.
Amortizing Loan Characteristics
- Definition: A loan where repayments include both interest and a portion of the principal amount.
Example Calculation: Monthly Loan Repayments
- Problem: Borrow $30,000 over 60 months at an APR of 6.75%, compounded monthly.
- Calculating inputs:
  r = \frac{0.0675}{12} = 0.005625 \quad n = 60 \quad PV = 30,000
- Monthly payment formula:
  C = \frac{PV}{\left(\frac{(1 + r)^{n} - 1}{r}\right)}
- Solve to find:
  C \approx 590.08

Market Forces in Interest Rate Determination
- Interest rates are shaped by relative supply and demand for funds.
- Interest Rate Formula:
  r = \text{real risk-free rate} + \text{inflation premium} + \text{risk premium}
Nominal vs Real Interest Rates
- Nominal Interest Rate: Represents the rate at which money will grow when invested.
- Real Interest Rate: Adjusted for inflation; reflects growth of purchasing power.
Fisher Equation
- The relation between nominal rates, real rates, and inflation:
  (1 + r) = (1 + r_{real})(1 + \text{inflation})
- Rearranged:
  r_{real} = \frac{(1 + r)}{(1 + \text{inflation})} - 1
Example of Real Interest Rate Calculation
- Given Data:
- 2005: Bond rates = 5.1%, Inflation = 2.7%
- 2016: Bond rates = 1.7%, Inflation = 1.3%
- Calculation of real interest rate for each year using Fisher Equation:
1. 2005: r_{real} = \frac{(1 + 0.051)}{(1 + 0.027)} - 1 \approx 2.34\%
2. 2016: r_{real} = \frac{(1 + 0.017)}{(1 + 0.013)} - 1 \approx 0.39\%
Conclusion
- Understanding interest rates involves recognizing their components, adjustments, and effects on financial decisions.
- Proper calculations enable informed decisions in investment, savings, and loan management.