15 Chapter 4 – Interest Rate
Answer Key for Conceptual Questions
1. Why Do Central Banks Adjust Interest Rates? Central banks, such as the Federal Reserve, adjust interest rates as a part of their monetary policy to control economic activity. Lowering rates encourages borrowing and investment, stimulating growth. Conversely, raising rates helps control inflation by reducing spending and investment.
2. How Does Inflation Impact Interest Rates? Inflation erodes the purchasing power of money over time, prompting lenders to demand higher nominal interest rates to compensate for this loss. Real interest rates, which account for inflation, represent the actual return on investment or cost of borrowing.
3. What Is the Yield Curve and Its Importance? The yield curve graphically represents the relationship between interest rates and the maturity of debt securities. A normal upward-sloping yield curve indicates expectations of economic growth, while an inverted curve can signal a potential recession. It’s a key tool for predicting economic trends.
4. Why Is the Opportunity Cost of Capital Important? The opportunity cost of capital reflects the return an investor foregoes by choosing one investment over another of similar risk. It serves as a benchmark for evaluating investment decisions, ensuring that resources are allocated to projects with returns exceeding this cost.
Short Calculations
1. Effective Annual Rate (EAR)
EAR = [latex](1 + \frac{\text{APR}}{n})^n - 1[/latex]
Given: APR = 18%, n = 12 (monthly compounding)
[latex]EAR = (1 + \frac{0.18}{12})^{12} - 1[/latex]
[latex]EAR = (1 + 0.015)^{12} - 1 = 1.19562 - 1 = 0.19562 = 19.56\%[/latex]
Answer: 19.56%
2. Comparing APR and EAR
EAR =[latex](1 + \frac{\text{APR}}{n})^n - 1[/latex]
Given: APR = 9.6%, n = 4 (quarterly compounding)
[latex]EAR = (1 + \frac{0.096}{4})^4 - 1[/latex]
[latex]EAR = (1 + 0.024)^4 - 1 = 1.09857 - 1 = 0.09857 = 9.86\%[/latex]
Answer: 9.86%
3. Real Interest Rate Calculation
Real Rate =[latex]\frac{1 + \text{Nominal Rate}}{1 + \text{Inflation Rate}} - 1[/latex]
Given: Nominal Rate = 7%, Inflation Rate = 3%
[latex]Real Rate = \frac{1 + 0.07}{1 + 0.03} - 1 = \frac{1.07}{1.03} - 1 = 1.03883 - 1 = 0.03883 = 3.88\%[/latex]
Answer: 3.88%
4. Loan Payment Calculation
Loan Payment Formula:
[latex]PMT = \frac{P \cdot r}{1 - (1 + r)^{-n}}[/latex]
Given: P = 15,000, APR = 5%, r = 0.05/12 = 0.004167, n = 4 \times 12 = 48
[latex]PMT = \frac{15,000 \cdot 0.004167}{1 - (1 + 0.004167)^{-48}}[/latex]
[latex]PMT = \frac{62.505}{1 - (1.004167)^{-48}} = \frac{62.505}{1 - 0.83565} = \frac{62.505}{0.16435} = 380.34[/latex]
Answer: $380.34
Scenario-Based Problems
1. Comparing Investment Options
Bank A:
[latex]EAR = (1 + \frac{0.05}{4})^4 - 1 = (1 + 0.0125)^4 - 1 = 1.05095 - 1 = 0.05095 = 5.10\%[/latex]
Bank B:
[latex]EAR = (1 + \frac{0.048}{12})^{12} - 1 = (1 + 0.004)^12 - 1 = 1.04901 - 1 = 0.04901 = 4.90\%[/latex]
Future Value (FV):
Bank A:
[latex]FV = 20,000 \times (1 + 0.0510)^3 = 20,000 \times 1.15763 = 23,152.60[/latex]
Bank B:
[latex]FV = 20,000 \times (1 + 0.0490)^3 = 20,000 \times 1.15392 = 23,078.40[/latex]
Answer: Bank A provides a slightly higher future value after 3 years.
2. Loan Comparison
Lender A:
[latex]PMT = \frac{30,000 \cdot 0.005}{1 - (1 + 0.005)^{-60}} = \frac{150}{1 - (1.005)^{-60}} = \frac{150}{0.25941} = 578.26[/latex]
Total Interest:
[latex]578.26 \times 60 - 30,000 = 4,695.60[/latex]
Lender B:
[latex]PMT = \frac{30,000 \cdot 0.004833}{1 - (1 + 0.004833)^{-60}} = \frac{144.99}{1 - (1.004833)^{-60}} = \frac{144.99}{0.25984} = 558.24[/latex]
Total Interest:
[latex]558.24 \times 60 - 30,000 = 3,494.40[/latex]
Answer: Lender B is more cost-effective with lower monthly payments and total interest.
3. Real vs. Nominal Returns
Real Return = [latex]\frac{1 + \text{Nominal Rate}}{1 + \text{Inflation Rate}} - 1[/latex]
[latex]Real Return = \frac{1 + 0.08}{1 + 0.03} - 1 = \frac{1.08}{1.03} - 1 = 0.04854 = 4.85\%[/latex]
Answer: 4.85%
Case Study: Understanding the Yield Curve
1. Interpretation:
An upward-sloping yield curve suggests that investors expect higher inflation and economic growth in the future, increasing the risk of long-term lending.
2. Cost of Borrowing:
5-Year Bonds:
[latex]1,000,000 \times (1 + 0.04)^5 = 1,216,653[/latex]
10-Year Bonds:
[latex]1,000,000 \times (1 + 0.05)^{10} = 1,628,895[/latex]
3. Inflation Expectations:
If inflation expectations rise, the company might lock in the lower 5-year rate. However, if long-term stability is valued, the 10-year option might be preferred despite higher costs.
Answer: Depends on risk tolerance and inflation outlook.
Interactive Challenge
1. Future Value (Option 1):
[latex]FV = 40,000 \times (1 + 0.06)^6 = 40,000 \times 1.41852 = 56,740.80[/latex]
2. Future Value (Option 2):
Using the FV annuity formula:
[latex]FV = PMT \times \frac{(1 + r)^n - 1}{r}[/latex]
[latex]FV = 600 \times \frac{(1 + 0.004167)^{72} - 1}{0.004167}[/latex]
[latex]FV = 600 \times \frac{1.34885 - 1}{0.004167} = 600 \times 83.684 = 50,210.40[/latex]
Answer: Both meet the goal, but Option 1 is more cost-effective.
