Financial Risk Manager Part 1

Explanation

This is a two-tailed test for testing whether the slope coefficient is different from 0. The confidence interval given is [0.30, 1.60] with a 10% critical value of 1.70.

Step 1: Calculate the slope coefficient (β̂)

The slope coefficient is the midpoint of the confidence interval: [ \hat{\beta} = \frac{0.30 + 1.60}{2} = 0.95 ]

Step 2: Calculate the standard error of β̂ (SEE_β̂)

Using the lower bound of the confidence interval: [ \hat{\beta} - C_t \times SEE_{\hat{\beta}} = 0.30 ] [ 0.95 - 1.70 \times SEE_{\hat{\beta}} = 0.30 ] [ 1.70 \times SEE_{\hat{\beta}} = 0.95 - 0.30 = 0.65 ] [ SEE_{\hat{\beta}} = \frac{0.65}{1.70} = 0.3824 ]

Step 3: Calculate the t-statistic

[ T = \frac{\hat{\beta} - \beta_{H_0}}{SEE_{\hat{\beta}}} = \frac{0.95 - 0}{0.3824} = 2.4843 ]

Step 4: Calculate the p-value

For a two-tailed test: [ p\text{-value} = 2[1 - \Phi(|T|)] = 2[1 - \Phi(2.4843)] ]

Using standard normal distribution tables:

• Φ(2.48) ≈ 0.9934
• Φ(2.49) ≈ 0.9936
• Interpolating for 2.4843: Φ(2.4843) ≈ 0.9935

[ p\text{-value} = 2[1 - 0.9935] = 2 \times 0.0065 = 0.0130 ]

This is very close to option A (0.0132), confirming that A is the correct answer.

Key points:

• This is a two-tailed test since we're testing if the slope is "different from 0"
• The confidence interval provides both the point estimate and standard error
• The p-value represents the probability of observing a test statistic as extreme as or more extreme than the calculated value, assuming the null hypothesis is true_