Let's break down the problem step by step: ### **Given:** - **Stock fund (S):** - Expected Return = **22%** - Standard Deviation = **32%** - **Bond fund (B):** - Expected Return = **12%** - Standard Deviation = **19%** - **Money Market fund (Risk-free rate, \( r_f \)) = 7%** - **Correlation between S and B, \( \rho_{SB} = 0.11 \)** - **Goal:** Find the Sharpe ratio of the **best feasible capital allocation line (CAL)**. --- ## **Step 1: Risky Portfolio with Maximum Sharpe Ratio** The **best feasible CAL** is achieved by combining the risky assets (S and B) in such a way that the Sharpe ratio is maximized. We first need to find the weights of S and B in the **tangency (optimal risky) portfolio**. Let: - \( E[R_S] = 0.22 \) - \( \sigma_S = 0.32 \) - \( E[R_B] = 0.12 \) - \( \sigma_B = 0.19 \) - \( r_f = 0.07 \) - \( \rho_{SB} = 0.11 \) The covariance: \[ \text{Cov}_{SB} = \rho_{SB} \cdot \sigma_S \cdot \sigma_B = 0.11 \times 0.32 \times 0.19 = 0.006688 \] Let \( w \) = weight in Stock fund, \( (1-w) \) = weight in Bond fund. Expected return of portfolio \( P \): \[ E[R_P] = w \cdot E[R_S] + (1-w) \cdot E[R_B] \] Variance of portfolio \( P \): \[ \sigma_P^2 = w^2 \sigma_S^2 + (1-w)^2 \sigma_B^2 + 2w(1-w)\text{Cov}_{SB} \] --- ## **Step 2: Find the Weights for the Tangency Portfolio** The weight in S (\( w^* \)) that maximizes the Sharpe ratio is: \[ w^* = \frac{E[R_S] - r_f}{\sigma_S^2} - \frac{E[R_B] - r_f}{\sigma_B^2} \cdot \frac{\text{Cov}_{SB}}{\sigma_B^2} \] But it's easier and more general to use the formula for two risky assets: \[ w^* = \frac{(E[R_S] - r_f) \sigma_B^2 - (E[R_B] - r_f) \text{Cov}_{SB}}{(\sigma_S^2)\sigma_B^2 - (\text{Cov}_{SB})^2} \] Let's calculate: - \( E[R_S] - r_f = 0.22 - 0.07 = 0.15 \) - \( E[R_B] - r_f = 0.12 - 0.07 = 0.05 \) - \( \sigma_S^2 = 0.1024 \) - \( \sigma_B^2 = 0.0361 \) - \( \text{Cov}_{SB} = 0.006688 \) Calculate denominator: \[ D = \sigma_S^2 \sigma_B^2 - (\text{Cov}_{SB})^2 = (0.1024)(0.0361) - (0.006688)^2 = 0.00369764 - 0.00004473 = 0.00365291 \] Calculate numerator: \[ N = (E[R_S] - r_f)\sigma_B^2 - (E[R_B] - r_f)\text{Cov}_{SB} = (0.15)(0.0361) - (0.05)(0.006688) = 0.005415 - 0.0003344 = 0.0050806 \] So, \[ w^* = \frac{0.0050806}{0.00365291} = 1.3910 \] Similarly, weight in B: \[ 1 - w^* = 1 - 1.3910 = -0.3910 \] --- ## **Step 3: Compute Expected Return and Standard Deviation of Tangency Portfolio** \[ E[R_P] = w^* \cdot E[R_S] + (1-w^*) \cdot E[R_B] = 1.3910 \times 0.22 + (-0.3910) \times 0.12 \] \[ = 0.30602 - 0.04692 = 0.2591 \] --- \[ \sigma_P^2 = (w^*)^2 \sigma_S^2 + (1-w^*)^2 \sigma_B^2 + 2 w^*(1-w^*) \text{Cov}_{SB} \] \[ (w^*)^2 = (1.3910)^2 = 1.9349 \] \[ (1-w^*)^2 = (-0.3910)^2 = 0.1529 \] \[ 2 w^*(1-w^*) = 2 \times 1.3910 \times (-0.3910) = -1.0878 \] So, \[ \sigma_P^2 = 1.9349 \times 0.1024 + 0.1529 \times 0.0361 + (-1.0878) \times 0.006688 \] \[ = 0.198179 + 0.005519 + (-0.007273) \] \[ = 0.198179 + 0.005519 - 0.007273 = 0.196425 \] \[ \sigma_P = \sqrt{0.196425} = 0.4434 \] --- ## **Step 4: Compute the Sharpe Ratio of the Tangency Portfolio** \[ \text{Sharpe Ratio} = \frac{E[R_P] - r_f}{\sigma_P} = \frac{0.2591 - 0.07}{0.4434} = \frac{0.1891}{0.4434} = 0.4267 \] --- ## **Final Answer** \[ \boxed{0.4267} \] --- **Summary of Steps:** 1. Calculate covariance and variances. 2. Compute weight in S for the tangency portfolio. 3. Find expected return and standard deviation for the tangency portfolio. 4. Calculate the Sharpe ratio. **Sharpe ratio of the best feasible CAL (rounded to 4 decimals) is:** \[ \boxed{0.4267} \]