Financial Risk Manager Part 1

Financial Risk Manager Part 1

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An analyst performed a regression of monthly returns on a stock with 4 independent variables over a 50 month period. The analyst calculated the total sum of squares (TSS) and the sum of square residuals or error (SSR) as 500 and 100, respectively. What is the adjusted R²?

TTanishq

Explanation:

Explanation

To calculate the adjusted R², we first need to find the regular R² and then apply the adjustment formula.

Step 1: Calculate R²

R2=1−SSRTSS=1−100500=1−0.2=0.8R^2 = 1 - \frac{SSR}{TSS} = 1 - \frac{100}{500} = 1 - 0.2 = 0.8

Where:

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

Step 2: Calculate Adjusted R²

Rˉ2=1−(n−1n−k−1)(1−R2)\bar{R}^2 = 1 - \left(\frac{n - 1}{n - k - 1}\right)(1 - R^2)

Where:

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

Rˉ2=1−(50−150−4−1)(1−0.8)=1−(4945)(0.2)=1−(1.0889×0.2)=1−0.2178=0.7822≈0.78\bar{R}^2 = 1 - \left(\frac{50 - 1}{50 - 4 - 1}\right)(1 - 0.8) = 1 - \left(\frac{49}{45}\right)(0.2) = 1 - (1.0889 \times 0.2) = 1 - 0.2178 = 0.7822 \approx 0.78

Why Adjusted R² is Lower

  • The adjusted R² penalizes for adding more independent variables
  • With 4 predictors and only 50 observations, there's a penalty for model complexity
  • The regular R² of 0.8 is adjusted downward to 0.78 to account for the number of predictors

Therefore, the correct answer is B: 0.78

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