Assuming that the expectations theory is the correct one of the term structure, calculate the interest rates in the term structure for maturities one to six years:
4%, 4.5%, 5%, 5.5%, 6%, 6.5%


To calculate the interest rates in the term structure according to the **Expectations Theory**, we use the idea that long-term interest rates are essentially an average of the current and expected future short-term rates. The formula for the \(n\)-year interest rate is:

\[
(1 + r_n)^n = (1 + r_1)(1 + r_2) \dots (1 + r_n)
\]

Where:
- \(r_n\) is the \(n\)-year interest rate.
- \(r_1, r_2, \dots\) are the rates for each year.

Given the rates: 
4% (1-year), 4.5% (2-year), 5% (3-year), 5.5% (4-year), 6% (5-year), and 6.5% (6-year), we need to apply this formula step by step to find the implied rates for each term structure.

### Step-by-step calculations:

1. **1-year rate (\(r_1\)):**
   \[
   r_1 = 4\% = 0.04
   \]

2. **2-year rate (\(r_2\)):**
   The two-year interest rate is the average of the 1-year rate and the 2nd year's implied rate (\(r_2\)):
   \[
   \left(1 + 0.045\right)^2 = (1 + 0.04)(1 + r_2)
   \]
   Solving for \(r_2\), we get:
   \[
   1.092025 = 1.04(1 + r_2)
   \]
   \[
   1 + r_2 = \frac{1.092025}{1.04} \approx 1.05
   \]
   \[
   r_2 = 0.05 \text{ or } 5\%
   \]

3. **3-year rate (\(r_3\)):**
   The three-year rate is the average of the 1-year, 2-year, and 3rd year's implied rate (\(r_3\)):
   \[
   \left(1 + 0.05\right)^3 = (1 + 0.04)(1 + 0.045)(1 + r_3)
   \]
   Solving for \(r_3\), we get:
   \[
   1.157625 = 1.04 \times 1.045 \times (1 + r_3)
   \]
   \[
   1 + r_3 = \frac{1.157625}{1.04 \times 1.045} \approx 1.06
   \]
   \[
   r_3 = 0.06 \text{ or } 6\%
   \]

4. **4-year rate (\(r_4\)):**
   Similarly:
   \[
   \left(1 + 0.055\right)^4 = (1 + 0.04)(1 + 0.045)(1 + 0.05)(1 + r_4)
   \]
   Solving for \(r_4\):
   \[
   1.240025 = 1.04 \times 1.045 \times 1.05 \times (1 + r_4)
   \]
   \[
   1 + r_4 = \frac{1.240025}{1.04 \times 1.045 \times 1.05} \approx 1.07
   \]
   \[
   r_4 = 0.07 \text{ or } 7\%
   \]

5. **5-year rate (\(r_5\)):**
   \[
   \left(1 + 0.06\right)^5 = (1 + 0.04)(1 + 0.045)(1 + 0.05)(1 + 0.055)(1 + r_5)
   \]
   Solving for \(r_5\):
   \[
   1.3401 = 1.04 \times 1.045 \times 1.05 \times 1.055 \times (1 + r_5)
   \]
   \[
   1 + r_5 = \frac{1.3401}{1.04 \times 1.045 \times 1.05 \times 1.055} \approx 1.075
   \]
   \[
   r_5 = 0.075 \text{ or } 7.5\%
   \]

6. **6-year rate (\(r_6\)):**
   \[
   \left(1 + 0.065\right)^6 = (1 + 0.04)(1 + 0.045)(1 + 0.05)(1 + 0.055)(1 + 0.06)(1 + r_6)
   \]
   Solving for \(r_6\):
   \[
   1.4756 = 1.04 \times 1.045 \times 1.05 \times 1.055 \times 1.06 \times (1 + r_6)
   \]
   \[
   1 + r_6 = \frac{1.4756}{1.04 \times 1.045 \times 1.05 \times 1.055 \times 1.06} \approx 1.08
   \]
   \[
   r_6 = 0.08 \text{ or } 8\%
   \]

### Summary of interest rates:

1. 1-year rate: 4%
2. 2-year rate: 5%
3. 3-year rate: 6%
4. 4-year rate: 7%
5. 5-year rate: 7.5%
6. 6-year rate: 8%

Let me know if you need further details or explanations.

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Here are 5 related questions you might be interested in:

1. How does the liquidity premium theory differ from expectations theory in explaining the term structure?
2. How would inflation expectations affect the term structure of interest rates?
3. Can you explain how forward rates are derived from spot rates?
4. What impact do central bank policies have on the term structure of interest rates?
5. How can businesses use the term structure to inform their borrowing decisions?

**Tip:** When dealing with bond pricing, knowing the term structure helps in predicting future interest rates and understanding the yield curve behavior.

Assuming that the expectations theory is the correct one of the term structure, calculate the interest rates in the term structure for maturities one to six years: 4%, 4.5%, 5%, 5.5%, 6%, 6.5%

Learn how to calculate interest rates for maturities from 1 to 6 years using Expectations Theory. Step-by-step calculations based on term structure rates of 4%, 4.5%, 5%, 5.5%, 6%, and 6.5%. This guide walks you through the mathematical formula and the implied rates for each year.

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