The unbiased expectations theory suggests that long-term interest rates reflect the expected future short-term rates. This theory can be used to calculate forward rates, which are the expected interest rates for future periods.

The formula to calculate the one-year forward rate beginning two years from today (denoted as $3f1$) using the unbiased expectations theory is:

$(1 + y_{3})^3 = (1 + y_{2})^2 (1 + 3f1)$

Where:

• $y_3$ is the 3-year yield (9.00% or 0.09).
• $y_2$ is the 2-year yield (6.50% or 0.065).
• $3f1$ is the 1-year forward rate for the period starting two years from now, which we need to find.

Step-by-Step Calculation:

1. Convert the yields to decimal form:

   • $y_3 = 0.09$
   • $y_2 = 0.065$

2. Substitute the known values into the equation:

$(1 + 0.09)^3 = (1 + 0.065)^2 (1 + 3f1)$

3. Simplify both sides:
   • $(1 + 0.09)^3 = 1.09^3 = 1.295029$
   • $(1 + 0.065)^2 = 1.065^2 = 1.134225$

$1.295029 = 1.134225 \times (1 + 3f1)$

4. Solve for $3f1$:

$\frac{1.295029}{1.134225} = 1 + 3f1$

$1.14185 = 1 + 3f1$

$3f1 = 0.14185 \quad \text{or} \quad 14.185\%$

Conclusion:

The one-year forward rate for the period beginning two years from today is approximately 14.185%.

Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. How does the unbiased expectations theory differ from the liquidity preference theory?
2. Can the forward rate be lower than the current spot rates?
3. How would different yield curve shapes affect forward rate calculations?
4. What real-world factors can cause deviations from the unbiased expectations theory?
5. How do changes in central bank policies impact forward rates?

Tip: Forward rates are useful tools for estimating future borrowing costs but can be affected by market expectations and risk premiums.