Let's carefully analyze and compute the covariance between each stock and the market index based on the information provided. --- ### Given Data: - **Stock A regression:** \[ R_A = 3.6\% + 1.20 R_M + e_A \] - **Stock B regression:** \[ R_B = -1.6\% + 1.50 R_M + e_B \] - **Market standard deviation:** \[ \sigma_M = 16\% \] - **R-squared values:** \[ R^2_A = 0.25, \quad R^2_B = 0.15 \] --- ### Step 1: Understand the relationship between R-squared, variance, and beta - In the index model: \[ R_i = \alpha_i + \beta_i R_M + e_i \] - The R-squared value represents the proportion of total variance in the stock's returns explained by the market: \[ R^2_i = \frac{\text{Variance explained by the market}}{\text{Total variance of stock i}} \] - The variance of stock i: \[ \sigma_{R_i}^2 = \text{Variance explained} + \text{variance of residual} \] - The variance explained by the market: \[ \beta_i^2 \sigma_M^2 \] - The residual variance: \[ \sigma_{e_i}^2 = (1 - R^2_i) \sigma_{R_i}^2 \] - **Note:** The covariance of stock i with the market is: \[ \text{Cov}(R_i, R_M) = \beta_i \sigma_M^2 \] - **Therefore,** to find the covariance, we need: - The beta (\(\beta_i\)) - The market variance (\(\sigma_M^2\)) --- ### Step 2: Extract \(\beta_i\) from the regression coefficients The regression coefficients provided are \(\beta_A = 1.20\) and \(\beta_B = 1.50\). --- ### Step 3: Calculate \(\sigma_M^2\) \[ \sigma_M = 16\% = 0.16 \] \[ \sigma_M^2 = (0.16)^2 = 0.0256 \] --- ### Step 4: Calculate the variance of each stock (\(\sigma_{R_i}^2\)) - For stock A: \[ R^2_A = 0.25 \] - Total variance of stock A: \[ \sigma_{R_A}^2 = \frac{\text{Variance explained}}{R^2_A} \] But more directly, the variance of stock A: \[ \sigma_{R_A}^2 = \text{variance explained} + \text{residual variance} \] Since: \[ R^2_A = \frac{\beta_A^2 \sigma_M^2}{\sigma_{R_A}^2} \] Rearranged: \[ \sigma_{R_A}^2 = \frac{\beta_A^2 \sigma_M^2}{R^2_A} \] Similarly for stock B: \[ \sigma_{R_B}^2 = \frac{\beta_B^2 \sigma_M^2}{R^2_B} \] --- ### Step 5: Calculate \(\sigma_{R_A}^2\) and \(\sigma_{R_B}^2\) - **Stock A:** \[ \sigma_{R_A}^2 = \frac{(1.20)^2 \times 0.0256}{0.25} \] \[ \sigma_{R_A}^2 = \frac{1.44 \times 0.0256}{0.25} \] \[ \sigma_{R_A}^2 = \frac{0.036864}{0.25} = 0.147456 \] - **Stock B:** \[ \sigma_{R_B}^2 = \frac{(1.50)^2 \times 0.0256}{0.15} \] \[ \sigma_{R_B}^2 = \frac{2.25 \times 0.0256}{0.15} \] \[ \sigma_{R_B}^2 = \frac{0.0576}{0.15} = 0.384 \] --- ### Step 6: Calculate covariance between each stock and the market \[ \text{Cov}(R_i, R_M) = \beta_i \sigma_M^2 \] - **Stock A:** \[ \text{Cov}(R_A, R_M) = 1.20 \times 0.0256 = 0.03072 \] - **Stock B:** \[ \text{Cov}(R_B, R_M) = 1.50 \times 0.0256 = 0.0384 \] --- ### **Final answers:** ### **Stock A:** \[ \boxed{ \text{Cov}(R_A, R_M) \approx 0.031 \text{ or } 3.1\% } \] ### **Stock B:** \[ \boxed{ \text{Cov}(R_B, R_M) \approx 0.038 \text{ or } 3.8\% } \] --- ### **Summary:** - **Stock A covariance with the market:** approximately **3%** - **Stock B covariance with the market:** approximately **4%** **Note:** Answers are rounded to the nearest whole number as instructed.