Answer: The expected short-term interest rate is 3.81% and the half-life is 11.6 years.

The correct answer is A: The expected short-term interest rate is 3.81% and the half-life is 11.6 years. The Vasicek model is a mean-reverting stochastic process used to model short-term interest rates. The equation for determining the expected short-term interest rate after a certain number of years (T) when the interest rate process follows a Vasicek model is given by: \[ r(T) = r_0 e^{-kT} + e \left(1 - e^{-kT}\right) \] Where: - \( r(T) \) is the expected short-term interest rate after time T. - \( r_0 \) is the current short-term interest rate. - \( e \) is the long-run value of the short-term interest rate. - \( k \) is the mean reversion rate. - \( T \) is the time in years. Using the provided information: - Current short-term interest rate \( r_0 = 3.35\% \) or 0.0335 in decimal form. - Long-run value \( e = 4.55\% \) or 0.0455 in decimal form. - Mean reversion rate \( k = 0.06 \). - Time \( T = 8 \) years. Plugging these values into the equation gives: \[ r(8) = 0.0335 e^{-0.06 \times 8} + 0.0455 \left(1 - e^{-0.06 \times 8}\right) \] \[ r(8) = 0.0335 \times 0.6188 + 0.0455 \times 0.3812 \] \[ r(8) = 0.0207 + 0.0173 \] \[ r(8) = 0.038 \text{ or } 3.81\% \] The half-life of the short-term interest rate, which is the time it takes for the rate to revert halfway to its long-run value, is given by the formula: \[ \text{Half-life} = \frac{\ln(2)}{k} \] Using the mean reversion rate \( k = 0.06 \): \[ \text{Half-life} = \frac{\ln(2)}{0.06} \approx 11.55 \text{ years} \] This calculation shows that the expected short-term interest rate after 8 years is approximately 3.81%, and the half-life of the interest rate is approximately 11.55 years, which rounds to 11.6 years when considering the options provided. Therefore, option A is the correct answer.

Author: LeetQuiz Editorial Team