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Answer: The expected short-term interest rate is 3.81% and the half-life is 11.6 years.
## Explanation ### Expected Short-Term Interest Rate Calculation For the Vasicek model, the expected short-term interest rate after time T is given by: $$ E[r_T] = r_0 \cdot e^{-kT} + \Theta \cdot (1 - e^{-kT}) $$ Given: - $r_0 = 3.35\% = 0.0335$ - $\Theta = 4.55\% = 0.0455$ - $k = 0.06$ - $T = 8$ years Calculation: $$ E[r_8] = 0.0335 \cdot e^{-0.06 \times 8} + 0.0455 \cdot (1 - e^{-0.06 \times 8}) $$ $$ E[r_8] = 0.0335 \cdot e^{-0.48} + 0.0455 \cdot (1 - e^{-0.48}) $$ $$ E[r_8] = 0.0335 \cdot 0.6188 + 0.0455 \cdot 0.3812 $$ $$ E[r_8] = 0.02073 + 0.01734 = 0.03807 = 3.81\% $$ ### Half-Life Calculation The half-life for mean reversion in the Vasicek model is: $$ \text{Half-life} = \frac{\ln(2)}{k} $$ Given $k = 0.06$: $$ \text{Half-life} = \frac{\ln(2)}{0.06} = \frac{0.6931}{0.06} = 11.55 \text{ years} \approx 11.6 \text{ years} $$ ### Why Other Options Are Incorrect - **Option B**: Uses incorrect half-life formula ($1/k = 16.7$ years instead of $\ln(2)/k$) - **Option C**: Incorrectly calculates expected rate as 4.09% instead of 3.81% - **Option D**: Contains both incorrect expected rate and incorrect half-life Therefore, the correct statement is that the expected short-term interest rate is 3.81% and the half-life is 11.6 years.
Author: LeetQuiz .
A quantitative analyst on the fixed-income desk of an investment bank is applying the Vasicek model to estimate future short-term interest rates. The model is given below:
where is the change in the short-term interest rate, is the estimated long-run value of the short-term interest rate, is the mean reversion rate, is the current level of the short-term interest rate, is the annual basis-point volatility of the short-term interest rate, is the time interval measured in years, and is a normally distributed random variable with mean zero and standard deviation equal to the square root of .
The analyst gathers the following information:
The analyst then creates an interest rate tree, determines the expected short-term interest rate after 8 years, and calculates how long it will take the short-term interest rate to revert halfway to the long-run value. Which of the following statements would be correct for the analyst to make?
A
The expected short-term interest rate is 3.81% and the half-life is 11.6 years.
B
The expected short-term interest rate is 3.81% and the half-life is 16.7 years.
C
The expected short-term interest rate is 4.09% and the half-life is 11.6 years.
D
The expected short-term interest rate is 4.09% and the half-life is 16.7 years.
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