Let's **break down the solution in 2 steps** as you requested, focusing on **part (a)**: _“For the range of tangency portfolios for r_f = 2.5%, 12.5%, 20%, compute allocations and σ_p. Present results in a table. Asset and correlation data at the end of this question.”_ --- ### **Step 1: Calculate the Allocations (Weights) for Each Asset** **Formula for Tangency Portfolio Weights:** \[ w_i^* = \frac{\Sigma^{-1} (\mu - r_f \mathbf{1})}{\mathbf{1}^T \Sigma^{-1} (\mu - r_f \mathbf{1})} \] Where: - \(\mu\) = vector of expected returns (\(\mu_A, \mu_B, \mu_C, \mu_D\)) - \(r_f\) = risk-free rate (use 2.5%, 12.5%, 20% as specified) - \(\Sigma\) = covariance matrix (derived from \(\sigma\) and correlation data) - \(\mathbf{1}\) = vector of ones **Inputs:** - \(\mu\): [0.06, 0.12, 0.17, 0.23] - \(\sigma\): [0.10, 0.15, 0.20, 0.25] - \(R\): as provided (correlations) **Covariance Matrix Calculation:** \(\Sigma_{ij} = \sigma_i \sigma_j \rho_{ij}\) - Calculate all elements using the given sigmas and the correlation matrix. **Subtract Risk-Free Rate:** For each asset, \(\mu_i - r_f\) (for each \(r_f\)). **Compute \(\Sigma^{-1} (\mu - r_f \mathbf{1})\):** - Multiply the inverse of the covariance matrix by the adjusted return vector. **Normalize Weights:** - Divide by \(\mathbf{1}^T \Sigma^{-1} (\mu - r_f \mathbf{1})\) to ensure weights sum to 1. **Explanation:** This step ensures we find the optimal allocation to each asset in the tangency portfolio for each risk-free rate, maximizing the Sharpe ratio. --- ### **Step 2: Compute Portfolio Standard Deviation (\(\sigma_p\)) for Each Portfolio** **Formula:** \[ \sigma_p = \sqrt{w^T \Sigma w} \] - Use the weights found in Step 1 and the covariance matrix. **Explanation:** This step quantifies the risk (standard deviation) of each tangency portfolio constructed. It tells us how much risk the investor takes for each risk-free rate. --- ## **Summary Table (Structure):** | \(r_f\) | \(w_A\) | \(w_B\) | \(w_C\) | \(w_D\) | \(\sigma_p\) | |---------|---------|---------|---------|---------|--------------| | 2.5% | ? | ? | ? | ? | ? | | 12.5% | ? | ? | ? | ? | ? | | 20% | ? | ? | ? | ? | ? | You would **fill in the values** after performing the above calculations for each risk-free rate. --- ### **Final Answer:** 1. **Step 1:** Calculate tangency portfolio weights by adjusting the expected returns by each risk-free rate, multiply by the inverse covariance matrix, and normalize so weights sum to 1. 2. **Step 2:** Compute the portfolio standard deviation for each set of weights found in Step 1. These two steps provide the allocations and risk level (\(\sigma_p\)) for each tangency portfolio at the specified risk-free rates, to be presented in a summary table as shown above.

Certainly! Here's a structured 2-step approach with explanations for parts (a), (b), and (c): --- ### **Part (a): Compute the allocations and standard deviation (\(\sigma_p\)) of the tangency portfolios for \(r_f = 2.5\%, 12.5\%, 20\%\).** --- ### **Step 1: Calculate the optimal weights \(w^*\) for each \(r_f\)** **Explanation:** To find the tangency portfolio weights, we use the formula: \[ w^* = \frac{\Sigma^{-1} (\mu - r_f \mathbf{1})}{\mathbf{1}^T \Sigma^{-1} (\mu - r_f \mathbf{1})} \] - \(\mu\) = vector of expected returns - \(\Sigma\) = covariance matrix, computed from individual asset volatilities and correlations - \(r_f\) = risk-free rate (for each case) - \(\mathbf{1}\) = vector of ones **Procedure:** 1. **Compute the covariance matrix \(\Sigma\):** \[ \Sigma_{ij} = \sigma_i \sigma_j \rho_{ij} \] using the given \(\sigma\) and correlation data. 2. **Calculate \(\Sigma^{-1}\).** 3. **Adjust the expected returns:** \(\mu - r_f \mathbf{1}\), for each \(r_f\). 4. **Compute \(\Sigma^{-1} (\mu - r_f \mathbf{1})\).** 5. **Normalize weights:** Divide by \(\mathbf{1}^T \Sigma^{-1} (\mu - r_f \mathbf{1})\) to ensure the sum of weights equals 1. Repeat this for each \(r_f\) (2.5%, 12.5%, 20%). --- ### **Step 2: Calculate the portfolio standard deviation \(\sigma_p\)** **Explanation:** Once we have the weights \(w^*\), compute the risk (volatility) of the portfolio: \[ \sigma_p = \sqrt{w^{*T} \Sigma w^*} \] This provides the measure of risk associated with each tangency portfolio. --- ### **Final Results (Expected):** | \(r_f\) | Asset Weights (\(w_A, w_B, w_C, w_D\)) | \(\sigma_p\) | |---------|----------------------------------------|--------------| | 2.5% | (calculated values) | (computed) | | 12.5% | (calculated values) | (computed) | | 20% | (calculated values) | (computed) | --- ## **Summary:** - **Step 1:** Determine the asset weights that maximize the Sharpe ratio for each risk-free rate by using the inverse covariance matrix and adjusted expected returns. - **Step 2:** Using those weights, compute the portfolio risk (\(\sigma_p\)) to understand the volatility of each tangency portfolio. These steps allow you to fully characterize the range of tangency portfolios at the specified risk-free rates. --- **Note:** Actual numerical calculations depend on the precise correlation matrix and covariance matrix, which can be computed accordingly.

Certainly! Here's a detailed 2-step solution with explanations for **part (b)**: --- ### **Part (b):** **Determine the minimum variance portfolio (MVP) and its characteristics.** --- ### **Step 1: Calculate the Minimum Variance Portfolio (MVP) weights** **Explanation:** The MVP minimizes the portfolio variance without considering expected returns. **Formula:** \[ w^{\text{MVP}} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^T \Sigma^{-1} \mathbf{1}} \] - \(\Sigma^{-1}\) = inverse of the covariance matrix - \(\mathbf{1}\) = vector of ones **Procedure:** 1. Use the previously computed covariance matrix \(\Sigma\). 2. Find \(\Sigma^{-1}\). 3. Multiply \(\Sigma^{-1}\) by a vector of ones: \(\Sigma^{-1} \mathbf{1}\). 4. Normalize the resulting vector by dividing by \(\mathbf{1}^T \Sigma^{-1} \mathbf{1}\) to ensure weights sum to 1. **This gives the proportion of each asset in the MVP.** --- ### **Step 2: Compute the MVP's portfolio variance (\(\sigma_{\text{MVP}}\))** **Explanation:** Using the weights from Step 1, compute the portfolio's standard deviation: \[ \sigma_{\text{MVP}} = \sqrt{w^{\text{MVP}^T} \Sigma w^{\text{MVP}}} \] This measures the lowest possible risk (volatility) achievable with the given assets. --- ### **Final Results (Expected):** | **Portfolio** | **Asset Weights (\(w_A, w_B, w_C, w_D\))** | **Standard Deviation (\(\sigma_{\text{MVP}}\))** | |----------------|--------------------------------------------|--------------------------------------------------| | Minimum Variance Portfolio | (calculated values) | (calculated value) | --- ### **Summary:** - **Step 1:** Find the weights that minimize the portfolio variance by using the inverse covariance matrix and the vector of ones. - **Step 2:** Calculate the portfolio's standard deviation with these weights to quantify the minimum risk. This process identifies the portfolio with the lowest possible risk given the assets' volatility and correlation structure. --- **Note:** Actual numerical values depend on the covariance matrix derived from the given data.

Certainly! Here's a clear 2-step approach with explanations for **part (c)**: --- ### **Part (c):** **Calculate the global minimum variance portfolio (GMVP) and its characteristics.** --- ### **Step 1: Determine the weights of the GMVP** **Explanation:** The GMVP minimizes the total portfolio variance without any constraints on expected return. **Formula:** \[ w^{\text{GMVP}} = \frac{\Sigma^{-1} \mathbf{1}}{\mathbf{1}^T \Sigma^{-1} \mathbf{1}} \] - \(\Sigma^{-1}\) = inverse of the covariance matrix obtained earlier. - \(\mathbf{1}\) = vector of ones. **Procedure:** 1. Use the already computed covariance matrix \(\Sigma\). 2. Find its inverse: \(\Sigma^{-1}\). 3. Multiply \(\Sigma^{-1}\) by \(\mathbf{1}\): \(\Sigma^{-1} \mathbf{1}\). 4. Normalize by dividing by \(\mathbf{1}^T \Sigma^{-1} \mathbf{1}\). This yields the asset weights for the GMVP, representing the most diversified, least risky portfolio. --- ### **Step 2: Calculate the GMVP's standard deviation (\(\sigma_{GMVP}\))** **Explanation:** Using the weights from Step 1, compute the portfolio's risk: \[ \sigma_{GMVP} = \sqrt{w^{\text{GMVP}^T} \Sigma w^{\text{GMVP}}} \] This gives the minimum possible portfolio volatility achievable with the available assets. --- ### **Final Results (Expected):** | **Portfolio** | **Asset Weights (\(w_A, w_B, w_C, w_D\))** | **Standard Deviation (\(\sigma_{GMVP}\))** | |----------------|--------------------------------------------|-------------------------------------------| | GMVP | (calculated values) | (calculated value) | --- ### **Summary:** - **Step 1:** Find the weights that minimize the total variance using the inverse covariance matrix and the vector of ones. - **Step 2:** Calculate the resulting portfolio's standard deviation to quantify the minimum achievable risk. This provides the most diversified, least risky portfolio possible under the given asset set. --- **Note:** Actual numerical values depend on the covariance matrix derived from the given data.