Let's break down the **portfolio optimization problem** and the **Lagrange multipliers** questions step by step using the information provided in the screenshots. --- ### **Step 1: Understanding the Model** #### **Variables** - \( x_1 = \) proportion in Stock - \( x_2 = \) proportion in Bond - \( x_3 = \) proportion in SCIP #### **Objective Function (Minimize Variance)** \[ \text{Minimize Variance} = 0.014697x_1^2 + 0.000155x_2^2 + 0.160791x_3^2 + 0.000936x_1x_2 - 0.004444x_1x_3 - 0.000454x_2x_3 \] #### **Constraints** 1. **Sum to 1:** \( x_1 + x_2 + x_3 = 1 \) 2. **Target Return:** \( 0.148x_1 + 0.06x_2 + 0.152x_3 \geq 0.12 \) 3. **Non-negativity:** \( x_1, x_2, x_3 \geq 0 \) --- ### **Step 2: Lagrange Multipliers Interpretation** - The **Lagrange multiplier for the total allocation** means how the objective (variance) would change with a small increase in the total allocation sum. But since \( x_1 + x_2 + x_3 = 1 \) is fixed, increasing the sum isn't meaningful. - The **Lagrange multiplier for the expected return constraint** tells you by how much the minimum variance would increase if the target return is increased by 1 unit (i.e., from 12% to 13%). --- ### **Step 3: Answering the Questions** #### **First Question: Lagrange multiplier for total allocation** - **Answer:** D. The Lagrange multiplier for the total allocation is meaningless because the total allocation must be 1. #### **Second Question: Lagrange multiplier for expected return allocation** - **Answer:** B. An increase in the target return from 12% to 13% would increase the variance by approximately **0.197**. This value comes from the solution shown, where 0.197 is given as the Lagrange multiplier for the expected return constraint. --- ### **Step 4: Boxed Final Answers** #### **First Question:** **D. The Lagrange multiplier for the total allocation is meaningless because the total allocation must be 1.** #### **Second Question:** **B. An increase in the target return from 12% to 13% would increase the variance by approximately 0.197.** --- ### **Explanation** - The model is a quadratic programming problem for portfolio optimization. - Lagrange multipliers give the rate of change of the objective function with respect to the right-hand side of constraints. - For the sum-to-one constraint, the Lagrange multiplier isn't meaningful since the sum must always be 1. - For the expected return constraint, the multiplier (0.197) shows the increase in variance for a 1.00 (or 100%) increase in target return. For a 1% (0.01) increase, multiply 0.197 by 0.01 to see the change. --- If you need help with any calculation or further interpretation, let me know!