Let's break down the problem and solve it step by step. ### **Given:** - \( w = 0.59 \) (fraction invested in stock fund) - \( 1-w = 0.41 \) (fraction invested in bond fund) - \( R = wR_s + (1-w)R_b \) - \( \mu_s = 0.09 \) (mean return of stock fund) - \( \sigma_s = 0.08 \) (std dev of stock fund) - \( \mu_b = 0.06 \) (mean return of bond fund) - \( \sigma_b = 0.05 \) (std dev of bond fund) - \( \rho = 0.29 \) (correlation between \( R_s \) and \( R_b \)) ### **Mean of \( R \):** \[ E[R] = wE[R_s] + (1-w)E[R_b] = w\mu_s + (1-w)\mu_b \] \[ E[R] = (0.59)(0.09) + (0.41)(0.06) \] \[ E[R] = 0.0531 + 0.0246 = 0.0777 \] --- ### **Variance of \( R \):** \[ \text{Var}(R) = w^2\sigma_s^2 + (1-w)^2\sigma_b^2 + 2w(1-w)\rho\sigma_s\sigma_b \] Plug in the values: \[ \text{Var}(R) = (0.59)^2(0.08^2) + (0.41)^2(0.05^2) + 2(0.59)(0.41)(0.29)(0.08)(0.05) \] Calculate each term: - \( (0.59)^2 = 0.3481 \) - \( (0.08)^2 = 0.0064 \) - \( (0.41)^2 = 0.1681 \) - \( (0.05)^2 = 0.0025 \) - \( 2 \times 0.59 \times 0.41 = 0.4838 \) - \( 0.29 \times 0.08 \times 0.05 = 0.00116 \) - \( 0.4838 \times 0.00116 = 0.000561208 \) Now sum the terms: - \( 0.3481 \times 0.0064 = 0.00222784 \) - \( 0.1681 \times 0.0025 = 0.00042025 \) - \( 0.000561208 \) Add them up: \[ \text{Var}(R) = 0.00222784 + 0.00042025 + 0.000561208 = 0.003209298 \] \[ \text{Std Dev}(R) = \sqrt{0.003209298} \approx 0.057 \] --- ### **Final Answers:** - **Mean:** \( \boxed{0.078} \) (rounded to three decimal places) - **Standard Deviation:** \( \boxed{0.057} \) (rounded to three decimal places) Let me know if you need any further clarifications!