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.