Financial Risk Manager Part 1

Explanation

Step 1: Calculate R²

R² (coefficient of determination) is calculated as:

$R^2 = 1 - \frac{SSR}{TSS}$

Where:

• SSR = Sum of Square Residuals (error) = 100
• TSS = Total Sum of Squares = 500

$R^2 = 1 - \frac{100}{500} = 1 - 0.2 = 0.8$

Step 2: Calculate Adjusted R²

The adjusted R² (denoted as $\bar{R}^2$) accounts for the number of independent variables and sample size. The formula is:

$\bar{R}^2 = 1 - \left(\frac{n - 1}{n - k - 1}\right)(1 - R^2)$

Where:

• n = number of observations = 50 months
• k = number of independent variables = 4
• R² = 0.8

Step 3: Plug in the values

$\bar{R}^2 = 1 - \left(\frac{50 - 1}{50 - 4 - 1}\right)(1 - 0.8)$

$\bar{R}^2 = 1 - \left(\frac{49}{45}\right)(0.2)$

$\bar{R}^2 = 1 - \frac{49}{45} \times 0.2$

$\bar{R}^2 = 1 - \frac{9.8}{45}$

$\bar{R}^2 = 1 - 0.2178$

$\bar{R}^2 = 0.7822 \approx 0.78$

Why Adjusted R² is lower than R²:

The adjusted R² penalizes the model for having additional independent variables that don't significantly improve the model's explanatory power. Since we have 4 independent variables and only 50 observations, the adjustment factor $\frac{n-1}{n-k-1} = \frac{49}{45} \approx 1.089$ reduces the R² value.

Key Formulas:

1. $R^2 = 1 - \frac{SSR}{TSS}`$2`. $\bar{R}^2 = 1 - \left(\frac{n-1}{n-k-1}\right)(1-R^2)$

Answer: B (0.78)