7. Predict the one-year interest rate two years from today if interest rates are 3.5%, 4.0%, 4.5%, and 5% for bonds with one to four years to maturity and liquidity premiums are 0%, 0.1%, 0.25%, and 0.50%.

To predict the one-year interest rate two years from today, we can use the **liquidity premium theory** of the term structure of interest rates. This theory accounts for both the expected future short-term interest rates and liquidity premiums (LP) when explaining the yield on long-term bonds.

Given:
- Interest rates for bonds of different maturities:
  - 1-year: 3.5%
  - 2-year: 4.0%
  - 3-year: 4.5%
  - 4-year: 5.0%
  
- Liquidity premiums for bonds of different maturities:
  - 1-year: 0.00%
  - 2-year: 0.10%
  - 3-year: 0.25%
  - 4-year: 0.50%

To find the one-year interest rate two years from today (\( i_{1,2} \)), we use the relationship:

\[
(1 + i_3)^3 = (1 + i_2)^2 \times (1 + i_{1,2})
\]

Where:
- \( i_3 \) is the observed 3-year bond yield (including the liquidity premium).
- \( i_2 \) is the observed 2-year bond yield (including the liquidity premium).
- \( i_{1,2} \) is the expected one-year interest rate two years from today (the value we're solving for).

### Step-by-Step Calculation:
1. **Remove the liquidity premiums** from the given bond yields to get the "pure" interest rates:
   - For the 3-year bond: 
     \[
     \text{Pure rate for 3-year bond} = 4.5\% - 0.25\% = 4.25\%
     \]
   - For the 2-year bond:
     \[
     \text{Pure rate for 2-year bond} = 4.0\% - 0.10\% = 3.90\%
     \]

2. **Convert the interest rates to decimal form** for easier calculations:
   - \( i_3 = 4.25\% = 0.0425 \)
   - \( i_2 = 3.90\% = 0.0390 \)

3. **Apply the formula**:

   Using the equation:
   \[
   (1 + 0.0425)^3 = (1 + 0.0390)^2 \times (1 + i_{1,2})
   \]

   First, calculate the left side:
   \[
   (1 + 0.0425)^3 = (1.0425)^3 \approx 1.1329
   \]

   Next, calculate \( (1 + 0.0390)^2 \):
   \[
   (1.0390)^2 \approx 1.0792
   \]

4. **Solve for \( i_{1,2} \)**:

   \[
   1.1329 = 1.0792 \times (1 + i_{1,2})
   \]
   Divide both sides by 1.0792:
   \[
   \frac{1.1329}{1.0792} \approx 1.0497
   \]
   So:
   \[
   1 + i_{1,2} = 1.0497 \quad \Rightarrow \quad i_{1,2} \approx 0.0497 \text{ or } 4.97\%
   \]

### Conclusion:
The expected one-year interest rate two years from today is approximately **4.97%**.

Would you like a more detailed explanation or any additional insights on this calculation?

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**Related Questions:**
1. How does the liquidity premium theory differ from the expectations theory?
2. How would the calculated rate change if the liquidity premium increased?
3. What would happen to the one-year forward rate if bond yields were inverted?
4. Can you use the same approach for predicting rates over longer periods?
5. How do interest rate expectations impact bond prices?

**Tip:** In bond yield calculations, adjusting for liquidity premiums is crucial, as they reflect compensation for holding longer-term bonds.

Predict the one-year interest rate two years from today if interest rates are 3.5%, 4.0%, 4.5%, and 5% for bonds with one to four years to maturity and liquidity premiums are 0%, 0.1%, 0.25%, and 0.50%.

Learn how to predict the one-year interest rate two years from today using liquidity premium theory. This detailed solution includes bond yields of 3.5%, 4.0%, 4.5%, and 5% for bonds maturing in one to four years, with liquidity premiums of 0%, 0.1%, 0.25%, and 0.50%. Suitable for college-level finance students.

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