Financial Risk Manager Part 2

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Explanation:

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.

Explanation:

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

$r(T) = r_0 e^{-kT} + e \left(1 - e^{-kT}\right)$

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 $k = 0.06$ :

$\text{Half-life} = \frac{\ln(2)}{0.06} \approx 11.55 \text{ years}$

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A fixed-income desk quantitative analyst at an investment bank aims to predict future short-term interest rates using the Vasicek model. This model is defined by the following equation: $dr = k * (e - r) * dt + g* dw$

In this context:

$dr$ represents the change in the short-term interest rate,

$e$ is the expected long-term average of the short-term interest rate,

$k$ denotes the speed of mean reversion,

$r$ is the current short-term interest rate,

$\sigma$ is the annual volatility of the short-term interest rate, quantified in basis points,

$dt$ signifies the time period in years, and

$dw$ is a random variable that follows a normal distribution with zero mean and a standard deviation equal to the square root of $dt$ .

The following data has been gathered for the analysis:

Current short-term interest rate ( $r$ ): 3.35%

Long-term expected short-term interest rate ( $e$ ): 4.55%

Mean reversion rate ( $k$ ): 0.06

Annual volatility in basis points ( $\sigma$ ): 120 bps

Using these inputs, the analyst constructs an interest rate tree and projects the anticipated short-term interest rate for the 8th year. Additionally, the analyst calculates the time required for the short-term interest rate to revert halfway to its long-term average. Based on this analysis, what would be an accurate statement for the analyst to make?

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

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