Answer: $ E(R_X) = 0.11 + 0.40 \times E(R_S) $

The correct answer is B: $ E(R_X) = 0.11 + 0.40 \times E(R_S) $ **Explanation:** 1. **Model Setup:** The linear regression model is given as: $$ R_X = a + b \times R_S + \epsilon_t $$ Taking expectations (since $E(\epsilon_t) = 0$): $$ E(R_X) = a + b \times E(R_S) $$ 2. **Calculating Slope Coefficient (b):** The slope coefficient in OLS regression is calculated as: $$ b = \frac{\text{Cov}(R_S, R_X)}{\text{Var}(R_S)} $$ Since correlation $\rho = \frac{\text{Cov}(R_S, R_X)}{\sigma_S \sigma_X}$, we can write: $$ \text{Cov}(R_S, R_X) = \rho \times \sigma_S \times \sigma_X $$ Therefore: $$ b = \frac{\rho \times \sigma_S \times \sigma_X}{\sigma_S^2} = \rho \times \frac{\sigma_X}{\sigma_S} $$ Substituting the given values: $$ b = 0.3 \times \frac{0.20}{0.15} = 0.3 \times 1.3333 = 0.40 $$ 3. **Calculating Intercept (a):** The intercept is calculated using the means: $$ a = \bar{R}_X - b \times \bar{R}_S $$ Substituting the given values: $$ a = 0.15 - 0.40 \times 0.10 = 0.15 - 0.04 = 0.11 $$ 4. **Final Model:** $$ E(R_X) = 0.11 + 0.40 \times E(R_S) $$ **Verification of Other Options:** - **Option A:** $0.40 + 0.40 \times E(R_S)$ - Incorrect intercept - **Option C:** $0.40 + 0.11 \times E(R_S)$ - Incorrect intercept and slope - **Option D:** None of the above - Incorrect since Option B is correct

Author: Nikitesh Somanthe