1. **Problem Statement:** Chris wants to allocate between two risky portfolios (S&P fund and Hedge fund) and a risk-free asset. We need to find the optimal risky allocation, expected return, and risk of the combined portfolio, and then the complete portfolio considering risk aversion. 2. **Given Data:** - S&P fund risk premium $\mu_1 = 0.05$, standard deviation $\sigma_1 = 0.20$ - Hedge fund risk premium $\mu_2 = 0.10$, standard deviation $\sigma_2 = 0.35$ - Risk-free rate $r_f = 0.01$ - Correlation $\rho = 0.3$ - Risk aversion $A = 3$ 3. **Formulas and Rules:** - Covariance $\sigma_{12} = \rho \sigma_1 \sigma_2$ - Covariance matrix $\Sigma = \begin{bmatrix} \sigma_1^2 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{bmatrix}$ - Expected returns vector $\mu = \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix}$ - Optimal risky portfolio weights $w^* = \frac{1}{A} \Sigma^{-1} \mu$ - Expected portfolio return $\mu_p = w^{*T} \mu$ - Portfolio variance $\sigma_p^2 = w^{*T} \Sigma w^*$ - Capital allocation to risky portfolio $y = \frac{\mu_p}{A \sigma_p^2}$ - Complete portfolio expected return $E[R_c] = r_f + y (\mu_p)$ - Complete portfolio standard deviation $\sigma_c = y \sigma_p$ 4. **Step-by-step Solution:** **(a) Optimal risky allocation $w^*$:** - Calculate covariance $\sigma_{12} = 0.3 \times 0.20 \times 0.35 = 0.021$ - Covariance matrix: $$\Sigma = \begin{bmatrix} 0.20^2 & 0.021 \\ 0.021 & 0.35^2 \end{bmatrix} = \begin{bmatrix} 0.04 & 0.021 \\ 0.021 & 0.1225 \end{bmatrix}$$ - Inverse of $\Sigma$: Calculate determinant: $$det = 0.04 \times 0.1225 - 0.021^2 = 0.0049 - 0.000441 = 0.004459$$ $$\Sigma^{-1} = \frac{1}{det} \begin{bmatrix} 0.1225 & -0.021 \\ -0.021 & 0.04 \end{bmatrix} = \begin{bmatrix} 27.46 & -4.71 \\ -4.71 & 8.97 \end{bmatrix}$$ - Expected returns vector: $$\mu = \begin{bmatrix} 0.05 \\ 0.10 \end{bmatrix}$$ - Compute $w^* = \frac{1}{A} \Sigma^{-1} \mu$ with $A=1$ for optimal risky portfolio weights (before considering risk aversion): $$w^* = \Sigma^{-1} \mu = \begin{bmatrix} 27.46 & -4.71 \\ -4.71 & 8.97 \end{bmatrix} \begin{bmatrix} 0.05 \\ 0.10 \end{bmatrix} = \begin{bmatrix} 27.46 \times 0.05 + (-4.71) \times 0.10 \\ -4.71 \times 0.05 + 8.97 \times 0.10 \end{bmatrix} = \begin{bmatrix} 1.373 - 0.471 \\ -0.2355 + 0.897 \end{bmatrix} = \begin{bmatrix} 0.902 \\ 0.6615 \end{bmatrix}$$ - Normalize weights to sum to 1: $$w_{sum} = 0.902 + 0.6615 = 1.5635$$ $$w = \begin{bmatrix} \frac{0.902}{1.5635} \\ \frac{0.6615}{1.5635} \end{bmatrix} = \begin{bmatrix} 0.577 \\ 0.423 \end{bmatrix}$$ **Answer (a):** Optimal risky allocation is approximately 57.7% in S&P fund and 42.3% in Hedge fund. **(b) Expected risk premium and standard deviation of portfolio:** - Expected risk premium: $$\mu_p = w^T \mu = 0.577 \times 0.05 + 0.423 \times 0.10 = 0.02885 + 0.0423 = 0.07115$$ - Portfolio variance: $$\sigma_p^2 = w^T \Sigma w = \begin{bmatrix} 0.577 & 0.423 \end{bmatrix} \begin{bmatrix} 0.04 & 0.021 \\ 0.021 & 0.1225 \end{bmatrix} \begin{bmatrix} 0.577 \\ 0.423 \end{bmatrix}$$ Calculate intermediate: $$\Sigma w = \begin{bmatrix} 0.04 \times 0.577 + 0.021 \times 0.423 \\ 0.021 \times 0.577 + 0.1225 \times 0.423 \end{bmatrix} = \begin{bmatrix} 0.02308 + 0.00888 \\ 0.01212 + 0.05182 \end{bmatrix} = \begin{bmatrix} 0.03196 \\ 0.06394 \end{bmatrix}$$ Then: $$w^T (\Sigma w) = 0.577 \times 0.03196 + 0.423 \times 0.06394 = 0.01844 + 0.02706 = 0.0455$$ - Standard deviation: $$\sigma_p = \sqrt{0.0455} = 0.2134$$ **Answer (b):** Expected risk premium is approximately 7.12% and standard deviation is approximately 21.3%. **(c) Capital allocation to risky portfolio and risk-free asset with $A=3$:** - Capital allocation to risky portfolio: $$y = \frac{\mu_p}{A \sigma_p^2} = \frac{0.07115}{3 \times 0.0455} = \frac{0.07115}{0.1365} = 0.521$$ - Capital allocation to risk-free asset: $$1 - y = 1 - 0.521 = 0.479$$ **Answer (c):** Chris should allocate approximately 52.1% to the risky portfolio and 47.9% to the risk-free asset. **(d) Expected return and standard deviation of complete portfolio:** - Expected return: $$E[R_c] = r_f + y \mu_p = 0.01 + 0.521 \times 0.07115 = 0.01 + 0.0371 = 0.0471$$ - Standard deviation: $$\sigma_c = y \sigma_p = 0.521 \times 0.2134 = 0.1112$$ **Answer (d):** Expected return is approximately 4.71% and standard deviation is approximately 11.1%.