Financial Risk Manager Part 1

Explanation

To test whether the slope coefficient is different from 0, we use a t-test with the following hypothesis:

• Null Hypothesis (H₀): β = 0
• Alternative Hypothesis (H₁): β ≠ 0

The test statistic is calculated as:

$T = \frac{\hat{\beta} - \beta_{H_0}}{\text{SEE}_{\hat{\beta}}}$

Where:

• $\hat{\beta} = 1.65$ (given slope coefficient)
• $\beta_{H_0} = 0$ (null hypothesis value)
• $\text{SEE}_{\hat{\beta}}$ is the standard error of the slope coefficient

Calculating Standard Error of $\hat{\beta}$

The standard error of the slope coefficient is:

$\text{SEE}_{\hat{\beta}} = \sqrt{\frac{s^2}{n \hat{\sigma}^2_X}}$

Given:

• $s^2 = 20.45$ (residual variance)
• $\hat{\sigma}^2_X = 18.65$ (variance of X)
• n = 120 (10 years × 12 months = 120 observations)

$\text{SEE}_{\hat{\beta}} = \sqrt{\frac{20.45}{120 \times 18.65}} = \sqrt{\frac{20.45}{2238}} = \sqrt{0.00914} = 0.0956$

Calculating Test Statistic

$T = \frac{1.65 - 0}{0.0956} = 17.2594$

Interpretation

The test statistic of 17.2594 is highly significant, indicating strong evidence against the null hypothesis that the slope coefficient equals zero. This suggests that the market return has a statistically significant relationship with the sales company's returns.